There are two major systems of units used in the world: SI units (acronym for the French Le Système International d’Unités, also known as the metric system), and English units (also known as the imperial system). English units were historically used in nations once ruled by the British Empire. Today, the United States is the only country that still uses English units extensively.

Virtually every other country in the world now uses the metric system, which is the standard system agreed upon by scientists and mathematicians.

Some physical quantities are more fundamental than others. In physics, there are seven fundamental physical quantities that are measured in base or physical fundamental units: length, mass, time, electric current temperature, amount of substance, and luminous intensity. Units for other physical quantities (such as force, speed, and electric charge) described by mathematically combining these seven base units. In this course, we will mainly use five of these: length, mass, time, electric current and temperature. The units in which they are measured are the meter, kilogram, second, ampere, kelvin, mole, and candela (Table 1.1). All other units are made by mathematically combining the fundamental units. These are called derived units.

Ǫuantity                 Name        Symbol

Length	Meter	m
Mass	Kilogram	kg
Time	Second	s
Electric current	Ampere	a
Temperature	Kelvin	k
Amount of substance	Mole	mol
Luminous intensity	Candela	cd

The Meter

The SI unit for length is the meter (m). The definition of the meter has changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum-iridium bar. (The bar is now housed at the International Bureau of Weights and Meaures, near Paris). By 1960, some distances could be measured more precisely by comparing them to wavelengths of light. The meter was redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition as the distance light travels in a vacuum in 1/ 299,792,458 of a second (Figure 1.14).

Figure 1.14 The meter is defined to be the distance light travels in 1/299,792,458 of a second through a vacuum. Distance traveled is speed multiplied by time.

The Kilogram

The SI unit for mass is the kilogram (kg). It is defined to be the mass of a platinum-iridium cylinder, housed at the International Bureau of Weights and Measures near Paris. Exact replicas of the standard kilogram cylinder are kept in numerous locations throughout the world, such as the National Institute of Standards and Technology in Gaithersburg, Maryland. The determination of all other masses can be done by comparing them with one of these standard kilograms.

The Second

The SI unit for time, the second (s) also has a long history. For many years it was defined as 1/86,400 of an average solar day. However, the average solar day is actually very gradually getting longer due to gradual slowing of Earth’s rotation. Accuracy in the fundamental units is essential, since all other measurements are derived from them. Therefore, a new standard was adopted to define the second in terms of a non-varying, or constant, physical phenomenon. One constant phenomenon is the very steady vibration of Cesium atoms, which can be observed and counted. This vibration forms the basis of the cesium atomic clock. In 1967, the second was redefined as the time required for 9,192,631,770 Cesium atom vibrations (Figure 1.15).

The Ampere

Electric current is measured in the ampere (A), named after Andre Ampere. You have probably heard of amperes, or amps, when people discuss electrical currents or electrical devices. Understanding an ampere requires a basic understanding of electricity and magnetism, something that will be explored in depth in later chapters of this book. Basically, two parallel wires with an electric current running through them will produce an attractive force on each other. One ampere is defined as the amount of electric current that will produce an attractive force of 2.7  10–7 newton per meter of separation between the two wires (the newton is the derived unit of force).                               

Kelvins

The SI unit of temperature is the kelvin (or kelvins, but not degrees kelvin). This scale is named after physicist William Thomson, Lord Kelvin, who was the first to call for an absolute temperature scale. The Kelvin scale is based on absolute zero. This is the point at which all thermal energy has been removed from all atoms or molecules in a system. This temperature, 0 K, is equal to −273.15 °C and −459.67 °F. Conveniently, the Kelvin scale actually changes in the same way as the Celsius scale. For example, the freezing point (0 °C) and boiling points of water (100 °C) are 100 degrees apart on the Celsius scale. These two temperatures are also 100 kelvins apart (freezing point = 273.15 K; boiling point = 373.15 K).

Metric Prefixes

Physical objects or phenomena may vary widely. For example, the size of objects varies from something very small (like an atom) to something very large (like a star). Yet the standard metric unit of length is the meter. So, the metric system includes many prefixes that can be attached to a unit. Each prefix is based on factors of 10 (10, 100, 1,000, etc., as well as 0.1, 0.01, 0.001, etc.). Table 1.2 gives the metric prefixes and symbols used to denote the different various factors of 10 in the metric system.

Prefix	Symbol	Value[1]	Example Name	Example Symbol	Example Value	Example Description
exa	E	1018	Exameter	Em	1018 m	Distance light travels in a century
peta	P	1015	Petasecond	Ps	1015 s	30 million years
tera	T	1012	Terawatt	TW	1012 W	Powerful laser output
giga	G	109	Gigahertz	GHz	109 Hz	A microwave frequency
mega	M	106	Megacurie	MCi	106 Ci	High radioactivity
kilo	k	103	Kilometer	km	103 m	About 6/10 mile
hector	h	102	Hectoliter	hL	102 L	26 gallons
deka	da	101	Dekagram	dag	101 g	Teaspoon of butter
deci	d	10–1	Deciliter	dL	10–1 L	Less than half a soda
centi	c	10–2	Centimeter	Cm	10–2 m	Fingertip thickness
milli	m	10–3	Millimeter	Mm	10–3 m	Flea at its shoulder
micro	µ	10–6	Micrometer	µm	10–6 m	Detail in microscope
nano	n	10–9	Nanogram	Ng	10–9 g	Small speck of dust
pico	p	10–12	Picofarad	pF	10–12 F	Small capacitor in radio
femto	f	10–15	Femtometer	Fm	10–15 m	Size of a proton
atto	a	10–18	Attosecond	as	10–18 s	Time light takes to cross an atom

The metric system is convenient because conversions between metric units can be done simply by moving the decimal place of a number. This is because the metric prefixes are sequential powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as U.S. customary units, the relationships are less simple—there are 12 inches in a foot, 5,280 feet in a mile, 4 quarts in a gallon, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by switching to the most-appropriate metric prefix. For example, distances in meters are suitable for building construction, but kilometers are used to describe road construction. Therefore, with the metric system, there is no need to invent new units when measuring very small or very large objects—you just have to move the decimal point (and use the appropriate prefix).

Known Ranges of Length, Mass, and Time

Table 1.3 lists known lengths, masses, and time measurements. You can see that scientists use a range of measurement units. This wide range demonstrates the vastness and complexity of the universe, as well as the breadth of phenomena physicists study. As you examine this table, note how the metric system allows us to discuss and compare an enormous range of phenomena, using one system of measurement (Figure 1.16 and Figure 1.17).

Length (m)	Phenomenon Measured	Mass (Kg)	Phenomenon Measured[1]	Time (s)	Phenomenon Measured[1]
10–18	Present experimental limit to smallest observable detail	10–30	Mass of an electron (9.11×10–31 kg)	10–23	Time for light to cross a proton
10–15	Diameter of a proton	10–27	Mass of a hydrogen atom (1.67×10–27 kg)	10–22	Mean life of an extremely unstable nucleus
1014	Diameter of a uranium nucleus	10–15	Mass of a bacterium	10–15	Time for one oscillation of a visible light
10–10	Diameter of a hydrogen atom	10–5	Mass of a mosquito	10–13	Time for one vibration of an atom in a solid
10–8	Thickness of membranes in cell of living organism	10–2	Mass of a hummingbird	10–8	Time for one oscillation of an FM radio wave
10–6	Wavelength of visible light	1	Mass of a liter of water (about a quart)	10–3	Duration of a nerve impulse
10–3	Size of a grain of sand	102	Mass of a person	1	Time for one heartbeat
1	Height of a 4-year-old child	103	Mass of a car	105	One day (8.64×104 s)
102	Length of a football field	108	Mass of a large ship	107	One day (3.16×107 s)
104	Greatest ocean depth	1012	Mass of a large iceberg	109	About half the life expectancy of a human
107	Diameter of Earth	1015	Mass of the nucleus of a comet	1011	Recorded history
1011	Distance from Earth to the sun	1023	Mass of the moon (7.35×1022 kg)	1017	Age of Earth
1016	Distance traveled by light in 1 year (a light year)	1025	Mass of Earth (5.97×1024kg)	1018	Age of the universe
1021	Diameter of the Milky Way Galaxy	1030	Mass of the Sun (1.99×1034 kg)
1022	Distance from Earth to the nearest large galaxy (Andromeda)	1042	Mass of the Milky Way galaxy (current upper limit)
1026	Distance from Earth to the edges of the known universe	1053	Mass of the known universe (current upper limit)

Using Scientific Notation with Physical Measurements

Scientific notation is a way of writing numbers that are too large or small to be conveniently written as a decimal. For example, consider the number 840,000,000,000,000. It’s a rather large number to write out. The scientific notation for this number is 8.40 x 10¹⁴. Scientific notation follows this general format

x x 10^y

In this format x is the value of the measurement with all placeholder zeros removed. In the example above, x is 8.4. The x is multiplied by a factor, 10y, which indicates the number of placeholder zeros in the measurement. Placeholder zeros are those at the end of a number that is 10 or greater, and at the beginning of a decimal number that is less than 1. In the example above, the factor is 1014. This tells you that you should move the decimal point 14 positions to the right, filling in placeholder zeros as you go. In this case, moving the decimal point 14 places creates only 13 placeholder zeros, indicating that the actual measurement value is 840,000,000,000,000.

Numbers that are fractions can be indicated by scientific notation as well. Consider the number 0.0000045. Its scientific notation is 4.5 x 10^–6. Its scientific notation has the same format

x x 10^y

Here, x is 4.5. However, the value of y in the 10y factor is negative, which indicates that the measurement is a fraction of 1. Therefore, we move the decimal place to the left, for a negative y. In our example of 4.5 x 10^–6, the decimal point would be moved to the left six times to yield the original number, which would be 0.0000045.

The term order of magnitude refers to the power of 10 when numbers are expressed in scientific notation. Quantities that have the same power of 10 when expressed in scientific notation, or come close to it, are said to be of the same order of magnitude. For example, the number 800 can be written as 8  102, and the number 450 can be written as 4.5  102. Both numbers have the same value for y. Therefore, 800 and 450 are of the same order of magnitude. Similarly, 101 and 99 would be regarded as the same order of magnitude, 102. Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of 10−9 m, while the diameter of the sun is on the order of 109 m. These two values are 18 orders of magnitude apart.

Scientists make frequent use of scientific notation because of the vast range of physical measurements possible in the universe, such as the distance from Earth to the moon (Figure 1.18), or to the nearest star.

Unit Conversion and Dimensional Analysis

It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook in the United States, some quantities may be expressed in liters and you need to convert them to cups. A Canadian tourist driving through the United States might want to convert miles to kilometers, to have a sense of how far away his next destination is. A doctor in the United States might convert a patient’s weight in pounds to kilograms.

Let’s consider a simple example of how to convert units within the metric system. How can we want to convert 1 hour to seconds?

Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. A conversion factor is simply a fraction which equals 1. You can multiply any number by 1 and get the same value. When you multiply a number by a conversion factor, you are simply multiplying it by one. For example, the following are conversion factors: (1 foot)/(12 inches) = 1 to convert inches to feet, (1 meter)/(100 centimeters) = 1 to convert centimeters to meters, (1 minute)/(60 seconds) = 1 to convert seconds to minutes. In this case, we know that there are 1,000 meters in 1 kilometer.

Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor (1 km/1,000m) = 1, so we are simply multiplying 80m by 1:

When there is a unit in the original number, and a unit in the denominator (bottom) of the conversion factor, the units cancel. In this case, hours and minutes cancel and the value in seconds remains.

You can use this method to convert between any types of unit, including between the U.S. customary system and metric system. Notice also that, although you can multiply and divide units algebraically, you cannot add or subtract different units. An expression like 10 km + 5 kg makes no sense. Even adding two lengths in different units, such as 10 km + 20 m does not make sense. You express both lengths in the same unit. See Appendix C for a more complete list of conversion factors.

Worked Example

Unit Conversions: A Short Drive Home

Suppose that you drive the 10.0 km from your university to home in 20.0 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s). (Note—Average speed is distance traveled divided by time of travel.)

Strategy

First we calculate the average speed using the given units. Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place.

Solution for (a)

1.Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.) In equation form,

2. Substitute the given values for distance and time.

3. Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is 60 min/1h . Thus

Discussion for (a)

To check your answer, consider the following:

1. Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel; rather, they will give you the wrong units as follows

which are obviously not the desired units of km/h.

2. Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of km/h and we have indeed obtained these units.

3. Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/h does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is defined to be 60 min, so the precision of the conversion factor is perfect.

4. Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable.

Worked Example

Using Physics to Evaluate Promotional Materials

A commemorative coin that is 2″ in diameter is advertised to be plated with 15 mg of gold. If the density of gold is 19.3 g/cc, and the amount of gold around the edge of the coin can be ignored, what is the thickness of the gold on the top and bottom faces of the coin?

Strategy

To solve this problem, the volume of the gold needs to be determined using the gold’s mass and density. Half of that volume is distributed on each face of the coin, and, for each face, the gold can be represented as a cylinder that is 2″ in diameter with a height equal to the thickness. Use the volume formula for a cylinder to determine the thickness.

Solution

Discussion

The amount of gold used is stated to be 15 mg, which is equivalent to a thickness of about 0.00019 mm. The mass figure may make the amount of gold sound larger, both because the number is much bigger (15 versus 0.00019), and because people may have a more intuitive feel for how much a millimeter is than for how much a milligram is. A simple analysis of this sort can clarify the significance of claims made by advertisers.

Accuracy, Precision and Significant Figures

Science is based on experimentation that requires good measurements. The validity of a measurement can be described in terms of its accuracy and its precision (see Figure 1.19 and Figure 1.20). Accuracy is how close a measurement is to the correct value for that measurement. For example, let us say that you are measuring the length of standard piece of printer paper. The packaging in which you purchased the paper states that it is 11 inches long, and suppose this stated value is correct. You measure the length of the paper three times and obtain the following measurements: 11.1 inches, 11.2 inches, and 10.9 inches. These measurements are quite accurate because they are very close to the correct value of 11.0 inches. In contrast, if you had obtained a measurement of 12 inches, your measurement would not be very accurate. This is why measuring instruments are calibrated based on a known measurement. If the instrument consistently returns the correct value of the known measurement, it is safe for use in finding unknown values.

Figure 1.19 A double-pan mechanical balance is used to compare different masses. Usually an object with unknown mass is placed in one pan and objects of known mass are placed in the other pan. When the bar that connects the two pans is horizontal, then the masses in both pans are equal. The known masses are typically metal cylinders of standard mass such as 1 gram, 10 grams, and 100 grams. (Serge Melki)

Figure 1.20 Whereas a mechanical balance may only read the mass of an object to the nearest tenth of a gram, some digital scales can measure the mass of an object up to the nearest thousandth of a gram. As in other measuring devices, the precision of a scale is limited to the last measured figures. This is the hundredths place in the scale pictured here. (Splarka, Wikimedia Commons)

Precision states how well repeated measurements of something generate the same or similar results. Therefore, the precision of measurements refers to how close together the measurements are when you measure the same thing several times. One way to analyze the precision of measurements would be to determine the range, or difference between the lowest and the highest measured values. In the case of the printer paper measurements, the lowest value was 10.9 inches and the highest value was 11.2 inches. Thus, the measured values deviated from each other by, at most, 0.3 inches. These measurements were reasonably precise because they varied by only a fraction of an inch. However, if the measured values had been 10.9 inches, 11.1 inches, and 11.9 inches, then the measurements would not be very precise because there is a lot of variation from one measurement to another.

The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull’s-eye target. Then think of each GPS attempt to locate the restaurant as a black dot on the bull’s eye.

In Figure 1.21, you can see that the GPS measurements are spread far apart from each other, but they are all relatively close to the actual location of the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system.

However, in Figure 1.22, the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system. Finally, in Figure 1.23, the GPS is both precise and accurate, allowing the restaurant to be located.

Figure 1.21 A GPS system attempts to locate a restaurant at the center of the bull’s-eye. The black dots represent each attempt to pinpoint the location of the restaurant. The dots are spread out quite far apart from one another, indicating low precision, but they are each rather close to the actual location of the restaurant, indicating high accuracy. (Dark Evil)

Figure 1.22 In this figure, the dots are concentrated close to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (Dark Evil)

Figure 1.23 In this figure, the dots are concentrated close to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (Dark Evil)

Uncertainty

The accuracy and precision of a measuring system determine the uncertainty of its measurements. Uncertainty is a way to describe how much your measured value deviates from the actual value that the object has. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values. For example, if someone asked you to provide the mileage on your car, you might say that it is 45,000 miles, plus or minus 500 miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the length of the paper is 11 inches plus or minus 0.2 inches or 11.0 ± 0.2 inches. The uncertainty in a

measurement, A, is often denoted as δA (“delta A“),

The factors contributing to uncertainty in a measurement include the following:

1. Limitations of the measuring device
2. The skill of the person making the measurement
3. Irregularities in the object being measured
4. Any other factors that affect the outcome (highly dependent on the situation)

In the printer paper example uncertainty could be caused by: the fact that the smallest division on the ruler is 0.1 inches, the person using the ruler has bad eyesight, or uncertainty caused by the paper cutting machine (e.g., one side of the paper is slightly longer than the other.) It is good practice to carefully consider all possible sources of uncertainty in a measurement and reduce or eliminate them,

Percent Uncertainty

One method of expressing uncertainty is as a percent of the measured value. If a measurement, A, is expressed with uncertainty,

Uncertainty in Calculations

There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the both the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements in the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor has a length of 4.00 m and a width of 3.00 m, with uncertainties of 2 percent and 1 percent, respectively, then the area of the floor is 12.0 m2 and has an uncertainty of 3 percent (expressed as an area this is 0.36 m2, which we round to 0.4 m2 since the area of the floor is given to a tenth of a square meter).

For a quick demonstration of the accuracy, precision, and uncertainty of measurements based upon the units of measurement, try this simulation (http://openstax.org/l/28precision) . You will have the opportunity to measure the length and weight of a desk, using milli- versus centi- units. Which do you think will provide greater accuracy, precision and uncertainty when measuring the desk and the notepad in the simulation? Consider how the nature of the hypothesis or research question might influence how precise of a measuring tool you need to collect data.

Precision of Measuring Tools and Significant Figures

An important factor in the accuracy and precision of measurements is the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, consider measuring the thickness of a coin. A standard ruler can measure thickness to the nearest millimeter, while a micrometer can measure the thickness to the nearest 0.005 millimeter. The micrometer is a more precise measuring tool because it can measure extremely small differences in thickness. The more precise the measuring tool, the more precise and accurate the measurements can be.

When we express measured values, we can only list as many digits as we initially measured with our measuring tool (such as the rulers shown in Figure 1.24). For example, if you use a standard ruler to measure the length of a stick, you may measure it with a decimeter ruler as 3.6 cm. You could not express this value as 3.65 cm because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36 mm and 37 mm. He or she must estimate the value of the last digit. The rule is that the last digit written down in a measurement is the first digit with some uncertainty. For example, the last measured value 36.5 mm has three digits, or three significant figures. The number of significant figures in a measurement indicates the precision of the measuring tool. The more precise a measuring tool is, the greater the number of significant figures it can report.

Figure 1.24 Three metric rulers are shown. The first ruler is in decimeters and can measure point three decimeters. The second ruler is in centimeters long and can measure three point six centimeters. The last ruler is in millimeters and can measure thirty-six point five millimeters.

Zeros

Special consideration is given to zeros when counting significant figures. For example, the zeros in 0.053 are not significant because they are only placeholders that locate the decimal point. There are two significant figures in 0.053—the 5 and the 3. However, if the zero occurs between other significant figures, the zeros are significant. For example, both zeros in 10.053 are significant, as these zeros were actually measured. Therefore, the 10.053 placeholder has five significant figures. The zeros in 1300 may or may not be significant, depending on the style of writing numbers. They could mean the number is known to the last zero, or the zeros could be placeholders. So 1300 could have two, three, or four significant figures. To avoid this ambiguity, write 1300 in scientific notation as 1.3 × 103. Only significant figures are given in the x factor for a number in scientific notation (in the form x x 10^y). Therefore, we know that 1 and 3 are the only significant digits in this number. In summary, zeros are significant except when they serve only as placeholders. Table 1.4 provides examples of the number of significant figures in various numbers.

Number	Significant Figures	Rationale
1.657	4	There are no zeros and all non-zero numbers are always significant.
0.4578	4	The first zero is only a placeholder for the decimal point.
0.000458	3	The first four zeros are placeholders needed to report the data to the ten-thoudsandths place.
2000.56	6	The three zeros are significant here because they occur between other significant figures.
45,600	3	With no underlines or scientific notation, we assume that the last two zeros are placeholders and are not significant.
15895000	7	The two underlined zeros are significant, while the last zero is not, as it is not underlined.
5.457 x 1013	4	In scientific notation, all numbers reported in front of the multiplication sign are significant
6.520 x 10–23	4	In scientific notation, all numbers reported in front of the multiplication sign are significant, including zeros.

Significant Figures in Calculations

When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and another rule for addition and subtraction, as discussed below.

1. For multiplication and division: The answer should have the same number of significant figures as the starting value with the fewest significant figures. For example, the area of a circle can be calculated from its radius using A = πr² . Let us see how many significant figures the area will have if the radius has only two significant figures, for example, r = 2.0 m. Then, using a calculator that keeps eight significant figures, you would get

A = πr² = (3.1415927…) x (2.0 m)² = 4.5238934 m²

But because the radius has only two significant figures, the area calculated is meaningful only to two significant figures or

A = 4.5 m²

even though the value of  is meaningful to at least eight digits.

• For addition and subtraction: The answer should have the same number places (e.g. tens place, ones place, tenths place, etc.) as the least-precise starting value. Suppose that you buy 7.56 kg of potatoes in a grocery store as measured with a scale having a precision of 0.01 kg. Then you drop off 6.052 kg of potatoes at your laboratory as measured by a scale with a precision of 0.001 kg. Finally, you go home and add 13.7 kg of potatoes as measured by a bathroom scale with a precision of 0.1 kg. How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer? The mass is found by simple addition and subtraction:

The least precise measurement is 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer should be rounded to the tenths place, giving 15.2 kg. The same is true for non-decimal numbers. For example,

6527.23 + 2 = 6528.23 = 6528

We cannot report the decimal places in the answer because 2 has no decimal places that would be significant. Therefore, we can only report to the ones place.

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Take Quiz

Most results in science are presented in scientific journal articles using graphs. Graphs present data in a way that is easy to visualize for humans in general, especially someone unfamiliar with what is being studied. They are also useful for presenting large amounts of data or data with complicated trends in an easily-readable way.

One commonly-used graph in physics and other sciences is the line graph, probably because it is the best graph for showing how one quantity changes in response to the other. Let’s build a line graph based on the data in Table 1.5, which shows the measured distance that a train travels from its station versus time. Our two variables, or things that change along the graph, are time in minutes, and distance from the station, in kilometers. Remember that measured data may not have perfect accuracy.

Time (min)  Distance from Station (km)

0	0
10	24
20	36
30	60
40	84
50	97
60	116
70	140

1. Draw the two axes. The horizontal axis, or x-axis, shows the independent variable, which is the variable that is controlled or manipulated. The vertical axis, or y-axis, shows the dependent variable, the non-manipulated variable that changes with (or is dependent on) the value of the independent variable. In the data above, time is the independent variable and should be plotted on the x-axis. Distance from the station is the dependent variable and should be plotted on the y-axis.

• Label each axes on the graph with the name of each variable, followed by the symbol for its units in parentheses. Be sure to leave room so that you can number each axis. In this example, use Time (min) as the label for the x-axis.
• Next, you must determine the best scale to use for numbering each axis. Because the time values on the x-axis are taken every 10 minutes, we could easily number the x-axis from 0 to 70 minutes with a tick mark every 10 minutes. Likewise, the y-axis scale should start low enough and continue high enough to include all of the distance from station values. A scale from 0 km to 160 km should suffice, perhaps with a tick mark every 10 km.

In general, you want to pick a scale for both axes that 1) shows all of your data, and 2) makes it easy to identify trends in your data. If you make your scale too large, it will be harder to see how your data change. Likewise, the smaller and more fine you make your scale, the more space you will need to make the graph. The number of significant figures in the axis values should be coarser than the number of significant figures in the measurements.

• Now that your axes are ready, you can begin plotting your data. For the first data point, count along the x-axis until you find the 10 min tick mark. Then, count up from that point to the 10 km tick mark on the y-axis, and approximate where 22 km is along the y-axis. Place a dot at this location. Repeat for the other six data points (Figure 1.26).

• Add a title to the top of the graph to state what the graph is describing, such as the y-axis parameter vs. the x-axis parameter. In the graph shown here, the title is train motion. It could also be titled distance of the train from the station vs. time.
• Finally, with data points now on the graph, you should draw a trend line (Figure 1.27). The trend line represents the dependence you think the graph represents, so that the person who looks at your graph can see how close it is to the real data. In the present case, since the data points look like they ought to fall on a straight line, you would draw a straight line as the trend line. Draw it to come closest to all the points. Real data may have some inaccuracies, and the plotted points may not all fall on the trend line. In some cases, none of the data points fall exactly on the trend line.

Analyzing a Graph Using Its Equation

One way to get a quick snapshot of a dataset is to look at the equation of its trend line. If the graph produces a straight line, the equation of the trend line takes the form

The b in the equation is the y-intercept while the m in the equation is the slope. The y-intercept tells you at what y value the line intersects the y-axis. In the case of the graph above, the y-intercept occurs at 0, at the very beginning of the graph. The y-intercept, therefore, lets you know immediately where on the y-axis the plot line begins.

The m in the equation is the slope. This value describes how much the line on the graph moves up or down on the y-axis along the line’s length. The slope is found using the following equation

In order to solve this equation, you need to pick two points on the line (preferably far apart on the line so the slope you calculate describes the line accurately). The quantities Y₂ and Y₁ represent the y-values from the two points on the line (not data points) that you picked, while X₂ and X₁ represent the two x-values of the those points.

What can the slope value tell you about the graph? The slope of a perfectly horizontal line will equal zero, while the slope of a perfectly vertical line will be undefined because you cannot divide by zero. A positive slope indicates that the line moves up the y-axis as the x-value increases while a negative slope means that the line moves down the y-axis. The more negative or positive the slope is, the steeper the line moves up or down, respectively. The slope of our graph in Figure 1.26 is calculated below based on the two endpoints of the line

Equation of line : y = (2.0 km/min) x + 0

Because the x axis is time in minutes, we would actually be more likely to use the time t as the independent (x-axis) variable and write the equation as

y = (2.0 km/min) t + 0

The formula y = mx + b only applies to linear relationships, or ones that produce a straight line. Another common type of line in physics is the quadratic relationship, which occurs when one of the variables is squared. One quadratic relationship in physics is the relation between the speed of an object its centripetal acceleration, which is used to determine the force needed to keep an object moving in a circle. Another common relationship in physics is the inverse relationship, in which one variable decreases whenever the other variable increases. An example in physics is Coulomb’s law. As the distance between two charged objects increases, the electrical force between the two charged objects decreases. Inverse proportionality, such the relation between x and y in the equation

y = k/x,

for some number k, is one particular kind of inverse relationship. A third commonly-seen relationship is the exponential relationship, in which a change in the independent variable produces a proportional change in the dependent variable. As the value of the dependent variable gets larger, its rate of growth also increases. For example, bacteria often reproduce at an exponential rate when grown under ideal conditions. As each generation passes, there are more and more bacteria to reproduce. As a result, the growth rate of the bacterial population increases every generation (Figure 1.28).

Using Logarithmic Scales in Graphing

Sometimes a variable can have a very large range of values. This presents a problem when you’re trying to figure out the best scale to use for your graph’s axes. One option is to use a logarithmic (log) scale. In a logarithmic scale, the value each mark labels

is the previous mark’s value multiplied by some constant. For a log base 10 scale, each mark labels a value that is 10 times the value of the mark before it. Therefore, a base 10 logarithmic scale would be numbered: 0, 10, 100, 1,000, etc. You can see how the logarithmic scale covers a much larger range of values than the corresponding linear scale, in which the marks would label the values 0, 10, 20, 30, and so on.

If you use a logarithmic scale on one axis of the graph and a linear scale on the other axis, you are using a semi-log plot. The Richter scale, which measures the strength of earthquakes, uses a semi-log plot. The degree of ground movement is plotted on a logarithmic scale against the assigned intensity level of the earthquake, which ranges linearly from 1-10 (Figure 1.29 (a)).

If a graph has both axes in a logarithmic scale, then it is referred to as a log-log plot. The relationship between the wavelength and frequency of electromagnetic radiation such as light is usually shown as a log-log plot (Figure 1.29 (b)). Log-log plots are also commonly used to describe exponential functions, such as radioactive decay.

Figure 1.29 (a) The Richter scale uses a log base 10 scale on its y-axis (microns of amplified maximum ground motion). (b) The relationship between the frequency and wavelength of electromagnetic radiation can be plotted as a straight line if a log-log plot is used.

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Take Quiz

Energy can be transferred to or from a system, either through a temperature difference between it andanother system (i.e., by heat) or by exerting a force through a distance (work). In these ways, energy can be converted into otherforms of energy in other systems. For example, a car engine burns fuel for heat transfer into a gas. Work is done by the gas as it exerts a force through a distance by pushing a piston outward. This work converts the energy into a variety of other forms—intoan increase in the car’s kinetic or gravitational potential energy; into electrical energy to run the spark plugs, radio, and lights;and back into stored energy in the car’s battery. But most of the thermal energy transferred by heat from the fuel burning in theengine does not do work on the gas. Instead, much of this energy is released into the surroundings at lower temperature (i.e.,lost through heat), which is quite inefficient. Car engines are only about 25 to 30 percent efficient. This inefficiency leads toincreased fuel costs, so there is great interest in improving fuel efficiency. However, it is common knowledge that moderngasoline engines cannot be made much more efficient. The same is true about the conversion to electrical energy in large powerstations, whether they are coal, oil, natural gas, or nuclear powered. Why is this the case?

The answer lies in the nature of heat. Basic physical laws govern how heat transfer for doing work takes place and limit themaximum possible efficiency of the process. This chapter will explore these laws as well their applications to everyday machines.

This chapter will explore these laws as well their applications to everyday machines. These topics are part of thermodynamics—the study of heat and its relationship to doing work.

Zeroth Law of Thermodynamics: Thermal Equilibrium

When two objects (or systems) are in contact with one another, heat will transfer thermal energy from the object at higher temperature to the one at lower temperature until they both reach the same temperature. The objects are then in thermal equilibrium, and no further temperature changes will occur if they are isolated from other systems. The systems interact and change because their temperatures are different, and the changes stop once their temperatures are the same. Thermal equilibrium is established when two bodies are in thermal contact with each other—meaning heat transfer (i.e., the transfer of energy by heat) can occur between them. If two systems cannot freely exchange energy, they will not reach thermal equilibrium. (It is fortunate that empty space stands between Earth and the sun, because a state of thermal equilibrium with the sun would be too toasty for life on this planet!)

If two systems, A and B, are in thermal equilibrium with each another, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C. This statement may seem obvious, because all three have the same temperature, but it is basic to thermodynamics. It is called the zeroth law of thermodynamics.

The zeroth law of thermodynamics is very similar to the transitive property of equality in mathematics: If a = b and b = c, then a = c

Pressure, Volume, Temperature, and the Ideal Gas Law

Before covering the first law of thermodynamics, it is first important to understand the relationship between pressure, volume, and temperature. Pressure, P, is defined as

P = F / A

where F is a force applied to an area, A, that is perpendicular to the force. Depending on the area over which it is exerted, a given force can have a significantly different effect

The SI unit for pressure is the pascal, where Pressure is defined for all states of matter but is particularly important when discussing fluids (such as air).

You have probably heard the word pressure being used in relation to blood (high or low blood pressure) and in relation to the weather (high- and low-pressure weather systems). These are only two of many examples of pressures in fluids.

The relationship between the pressure, volume, and temperature for an ideal gas is given by the ideal gas law. A gas is considered ideal at low pressure and fairly high temperature, and forces between its component particles can be ignored.

The ideal gas law states that

PV = NkT

where P is the pressure of a gas, V is the volume it occupies, N is the number of particles (atoms or molecules) in the gas, , and T is its absolute temperature. The constant k is called the Boltzmann constant and has the value ( k = 1.38 X 10^-23 J/K)

It is important for us to notice from the equation that the following are true for a given mass of gas:

• When volume is constant, pressure is directly proportional to temperature.

 • When temperature is constant, pressure is inversely proportional to volume.

• When pressure is constant, volume is directly proportional to temperature.

This last point describes thermal expansion—the change in size or volume of a given mass with temperature. What is the underlying cause of thermal expansion? An increase in temperature means that there’s an increase in the kinetic energy of the individual atoms. Gases are especially affected by thermal expansion, although liquids expand to a lesser extent with similar increases in temperature, and even solids have minor expansions at higher temperatures. This is why railroad tracks and bridges have expansion joints that allow them to freely expand and contract with temperature changes.

To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into a deflated tire. The tire’s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If you continue to pump air into tire (which now has a nearly constant volume), the pressure increases with increasing temperature)

(a) When air is pumped into a deflated tire, its volume first increases without much increase in pressure.

(b) When the tire is filled to a certain point, the tire walls resist further expansion, and the pressure increases as more air is added. (

c) Once the tire is inflated fully, its pressure increases with temperature.

The First Law of Thermodynamics :

 Heat (Q) and work (W) are the two ways to add or remove energy from a system. The processes are very different. Heat is driven by temperature differences, while work involves a force exerted through a distance. Nevertheless, heat and work can produce identical results. For example, both can cause a temperature increase. Heat transfers energy into a system, such as when the sun warms the air in a bicycle tire and increases the air’s temperature. Similarly, work can be done on the system, as when the bicyclist pumps air into the tire. Once the temperature increase has occurred, it is impossible to tell whether it was caused by heat or work. Heat and work are both energy in transit—neither is stored as such in a system. However, both can change the internal energy, U, of a system.

Internal energy is the sum of the kinetic and potential energies of a system’s atoms and molecules. It can be divided into many subcategories, such as thermal and chemical energy, and depends only on the state of a system (that is, P, V, and T), not on how the energy enters or leaves the system.

In order to understand the relationship between heat, work, and internal energy, we use the first law of thermodynamics. The first law of thermodynamics applies the conservation of energy principle to systems where heat and work are the methods of transferring energy into and out of the systems. It can also be used to describe how energy transferred by heat is converted and transferred again by work.

The first law of thermodynamics states that the change in internal energy of a closed system equals the net heat transfer into the system minus the net work done by the system. In equation form, the first law of thermodynamics is

ΔU = Q – W

Here, ΔU  is the change in internal energy, U, of the system. As shown in Figure 12.6, Q is the net heat transferred into the system—that is, Q is the sum of all heat transfers into and out of the system. W is the net work done by the system—that is, W is the sum of all work done on or by the system. By convention, if Q is positive, then there is a net heat transfer into the system; if W is positive, then there is net work done by the system. So positive Q adds energy to the system by heat, and positive W takes energy from the system by work. Note that if heat transfers more energy into the system than that which is done by work, the difference is stored as internal energy

The first law of thermodynamics is the conservation of energy principle stated for a system, where heat and work are the methods of transferring energy to and from a system. Q represents the net heat transfer—it is the sum of all transfers of energy by heat into and out of the system. Q is positive for net heat transfer into the system. W_out is the work done by the system, and W_in is the work done on the system. W is the total work done on or by the system. W is positive when more work is done by the system than on it. The change in the internal energy of the system, ΔU , is related to heat and work by the first law of thermodynamics: ΔU = Q – W

It follows also that negative Q indicates that energy is transferred away from the system by heat and so decreases the system’s internal energy, whereas negative W is work done on the system, which increases the internal energy.

Second Law of Thermodynamics: Entropy

It is not even theoretically possible for engines to be 100 percent efficient. This phenomenon is explained by the second law of thermodynamics, which relies on a concept known as entropy. Entropy is a measure of the disorder of a system. Entropy also describes how much energy is not available to do work. The more disordered a system and higher the entropy, the less of a system’s energy is available to do work.

Although all forms of energy can be used to do work, it is not possible to use the entire available energy for work. Consequently, not all energy transferred by heat can be converted into work, and some of it is lost in the form of waste heat—that is, heat that does not go toward doing work. The unavailability of energy is important in thermodynamics; in fact, the field originated from efforts to convert heat to work, as is done by engines.

The equation for the change in entropy, ΔS  , is ΔS = Q/T

where Q is the heat that transfers energy during a process, and T is the absolute temperature at which the process takes place. Q is positive for energy transferred into the system by heat and negative for energy transferred out of the system by heat. In SI, entropy is expressed in units of joules per kelvin (J/K). If temperature changes during the process, then it is usually a good approximation (for small changes in temperature) to take T to be the average temperature

The second law of thermodynamics states that the total entropy of a system either increases or remains constant in any spontaneous process; it never decreases.

An important implication of this law is that heat transfers energy spontaneously from higher- to lower-temperature objects, but never spontaneously in the reverse direction. This is because entropy increases for heat transfer of energy from hot to cold. Because the change in entropy is Q/T, there is a larger change in at lower temperatures (smaller T). The decrease in entropy of the hot (larger T) object is therefore less than the increase in entropy of the cold (smaller T) object, producing an overall increase in entropy for the system

The ice in this drink is slowly melting. Eventually, the components of the liquid will reach thermal equilibrium, as predicted by the second law of thermodynamics—that is, after heat transfers energy from the warmer liquid to the colder ice

Another way of thinking about this is that it is impossible for any process to have, as its sole result, heat transferring energy from a cooler to a hotter object. Heat cannot transfer energy spontaneously from colder to hotter, because the entropy of the overall system would decrease.

Suppose we mix equal masses of water that are originally at two different temperatures, say and . The result will be water at an intermediate temperature of 30⁰ C . Three outcomes have resulted: entropy has increased, some energy has become unavailable to do work, and the system has become less orderly.

Let us think about each of these results. First, why has entropy increased? Mixing the two bodies of water has the same effect as the heat transfer of energy from the higher-temperature substance to the lower-temperature substance. The mixing decreases the entropy of the hotter water but increases the entropy of the colder water by a greater amount, producing an overall increase in entropy.

Second, once the two masses of water are mixed, there is no more temperature difference left to drive energy transfer by heat and therefore to do work. The energy is still in the water, but it is now unavailable to do work.

Third, the mixture is less orderly, or to use another term, less structured. Rather than having two masses at different temperatures and with different distributions of molecular speeds, we now have a single mass with a broad distribution of molecular speeds, the average of which yields an intermediate temperature. These three results—entropy, unavailability of energy, and disorder—not only are related but are, in fact, essentially equivalent. Heat transfer of energy from hot to cold is related to the tendency in nature for systems to become disordered and for less energy to be available for use as work.

Based on this law, what cannot happen? A cold object in contact with a hot one never spontaneously transfers energy by heat to the hot object, getting colder while the hot object gets hotter. Nor does a hot, stationary automobile ever spontaneously cool off and start moving

We’ve explained that heat never transfers energy spontaneously from a colder to a hotter object. The key word here is spontaneously. If we do work on a system, it is possible to transfer energy by heat from a colder to hotter object. We’ll learn more about this in the next section, covering refrigerators as one of the applications of the laws of thermodynamics. Sometimes people misunderstand the second law of thermodynamics, thinking that based on this law, it is impossible for entropy to decrease at any particular location. But, it actually is possible for the entropy of one part of the universe to decrease, as long as the total change in entropy of the universe increases. In equation form, we can write this as

ΔS_tot = ΔS_system + ΔS_environment  > 0

Based on this equation, we see that ΔS_system can be negative as long as ΔS_environment    is positive and greater in magnitude.

How is it possible for the entropy of a system to decrease? Energy transfer is necessary.

• If you pick up marbles that are scattered about the room and put them into a cup, your work has decreased the entropy of that system.
• If you gather iron ore from the ground and convert it into steel and build a bridge, your work has decreased the entropy of that system. . In this case , although you made the system of the bridge and steel more structured, you did so at the expense of the universe. Altogether, the entropy of the universe is increased by the disorder created by digging up the ore and converting it to stee
• Energy coming from the sun can decrease the entropy of local systems on Earth—that is, is negative. But the overall entropy of the rest of the universe increases by a greater amount—that is, is positive and greater in magnitude

Take Quiz

Kinematics is defined as the study of motion without considering its causes. The word “kinematics” comes from a Greek term meaning motion and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). In one-dimensional kinematics we will study only the motion of a ball, for example, without worrying about what forces cause or change its motion. Such considerations come in other chapters. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one-dimensional motion.

Position

In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the positionof the rocket with respect to the Earth as a whole, while a professor’s positioncould be described in terms of where she is in relation to the nearby white board.  In other cases, we use reference frames that are not stationary but are in motionrelative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame.  

Displacement

If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object’s position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced.

Definition: 

DISPLACEMENT

Displacement is the change in position of an object:

Δx=xf−x0,

where Δx is displacement , x_f is the final position and x ₀ is the initial position

In this text the upper case Greek letter Δ always means “change in” whatever quantity follows it; thus, Δx means change in position

We can find displacement by subtracting initial position X₀ from the final position X_f

Note that the SI unit for displacement is the meter (m), but sometimes kilometres, miles, feet, and other units of length are used. Keep in mind that when units other than the meter are used in a problem, you may need to convert them into meters to complete the calculation.

A professor paces left and right while lecturing. Her position relative to Earth is given by x. The +2.0 m displacement of the professor relative to Earth is represented by an arrow pointing to the right.

A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by x. The −4.0 m displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far)  

Note that displacement has a direction as well as a magnitude. The professor’s displacement is 2.0 m to the right, and the airline passenger’s displacement is 4.0 m toward the rear. In one-dimensional motion, direction can be specified with a plus or minus sign. When you begin a problem, you should select which direction is positive (usually that will be to the right or up, but you are free to select positive as being any direction).

The professor’s initial position is X₀ =1.5 m and her final position is X_f =3.5 m . Thus her displacement is

Δx=Xf−X0=3.5 m−1.5 m=+2.0 m.

In this coordinate system, motion to the right is positive, whereas motion to the left is negative.

Similarly, the airplane passenger’s initial position is X0=6.0 m and his final position is Xf=2.0m and hence his displacement is

Δx=Xf−X0=2.0 m−6.0 m=−4.0 m.

His displacement is negative because his motion is toward the rear of the plane, or in the negative x direction in our coordinate system.

Distance

Although displacement is described in terms of direction, distance is not. 

Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the 

distance travelled between them. 

Distance travelled is the total length of the path traveled between two positions. 

Distancehas no direction and, thus, no sign. For example, the distance

 the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m.

MISCONCEPTION ALERT: 

DISTANCE TRAVELED VS. MAGNITUDE OF DISPLACEMENT

It is important to note that the distance traveled, however, can be greater than the magnitude of the displacement (by magnitude, we mean just the size of the displacement without regard to its direction; that is, just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 m during a lecture, yet still end up only 2.0 m to the right of her starting point. In this case her displacement would be +2.0 m, the magnitude of her displacement would be 2.0 m, but the distance she travelled would be 150 m.

In kinematics we nearly always deal with displacement and magnitude of displacement, and almost never with distance travelled. One way to think about this is to assume you marked the start of the motion and the end of the motion. The 

Displacement is simply the difference in the position of the two marks and is independent of the path taken in traveling between the two marks. The 

distance travelled, however, is the total length of the path taken between the two marks.

Exercise 

A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is her displacement? (b) What distance does she ride? (c) What is the magnitude of her displacement?

Answer

The rider’s displacement is .

a) Δx=Xf−X0 = 2 – 3 = -1  (The displacement is negative because we take east to be positive and west to be negative.)

 (b) The distance traveled is 3 km + 2 km = 5 km.

(c) The magnitude of the displacement is 1 km

Vectors, Scalars, and Coordinate Systems

What is the difference between distance and displacement?

Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity.

A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down. The direction of a vector in one-dimensional motion is given simply by a plus (+) or minus (−) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector’s magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector. Some physical quantities, like distance, either have no direction or none is specified.

A scalar is any quantity that has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km/h speed limit, a person’s 1.8 m height, and a distance of 2.0 m are all scalars—quantities with no specified direction. Note, however, that a scalar can be negative, such as a −20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows.

Coordinate Systems for One-Dimensional Motion

In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in Figure , it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it. Figure :

It is usually convenient to consider motion upward or to the right as positive (+) and motion downward or to the left as negative (−).

Time, Velocity, and Speed :

There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we add definitions of time, velocity, and speed to expand our description of motion.

Time

The most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple—time is change, or the interval over which change occurs. It is impossible to know that time has passed unless something changes. The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us to not only measure the amount of time, but also to determine a sequence of events.

How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M., so that the elapsed time would be 50 min.

Elapsed time Δt is the difference between the ending time and beginning time,

Δt = t_f – t₀ , where

Where Δt is the change in time or elapsed time, is the time at the end of the motion, and t₀ is the time at the beginning of the motion.

Velocity

Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has units of distance divided by time, such as miles per hour or kilometres per hour.

Average velocity is displacement (change in position) divided by the time of travel,

ṽ =  Δx / Δt  = (X_f −X₀) /  (t_f −t₀)

where ṽ  is the average (indicated by the bar over the v) velocity, Δx is the change in position (or displacement), and X_f  and X₀ are the final and beginning positions at times t_f and t₀  respectively

Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity is meters per second or m/s, but many other units, such as km/h, mi/h (also written as mph), and cm/s, are in common use.

Suppose, for example, an airplane passenger took 5 seconds to move −4 m (the negative sign indicates that displacement is toward the back of the plane). His average velocity would be

ṽ =  Δx / Δt  = (-4m)/5s = -0.8 m/s

The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he goes to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals.

The smaller the time intervals considered in a motion, the more detailed the information. When we carry this process to its logical conclusion, we are left with an infinitesimally small interval. Over such an interval, the average velocity becomes the instantaneous velocity or the velocity at a specific instant. A car’s speedometer, for example, shows the magnitude (but not the direction) of the instantaneous velocityof the car. (Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to use average velocity.) 

Instantaneous velocity v is the average velocity at a specific instant in time (or over an infinitesimally small timeinterval).

Speed

In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is that speed has no direction. Thus speed is a scalar . Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed

Instantaneous speed is the magnitude of instantaneous velocity . For example, suppose the airplane passenger at one instant had an instantaneous velocity  of −3.0 m/s (the minus meaning toward the rear of the plane). At that same time  his instantaneous speed was 3.0 m/s. Or suppose that at one time during a shopping trip your instantaneous velocity is 40 km/h due north. Your instantaneous speed at that instant would be 40 km/h—the same magnitude but without a direction. 

Average speed , however, is very different from average velocity.  Average speed  is the distance travelled  divided by elapsed time .

We have noted that distance travelled  can be greater than displacement. So average speedcan be greater than average velocity , which is displacement  divided by time . For example, if you drive to a store and return home in half an hour, and your car’s odometer shows the total distance travelled  was 6 km, then your average speed  was 12 km/h. Your average velocity , however, was zero, because your displacement  for the round trip is zero. (Displacement is change in position  and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity

Another way of visualizing the motion  of an object is to use a graph. A plot of position or of velocity as a function of time  can be very useful. For example, for this trip to the store, the position , velocity, and speed-vs.-time graphs are displayed (Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we’ll probably stop at the store. But for simplicity’s sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.)

.

Position vs. time , velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative.

Acceleration :

In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time . The formal definition of 

Acceleration is consistent with these notions, but more inclusive.

AVERAGE ACCELERATION

Average Acceleration is the rate at which velocity changes,

ặ=Δv / Δt=(V_f−V₀) / (T_f−T₀)

where ặ  is average acceleration ,V is velocity, and T is time

Because acceleration is velocity in m/s divided by time  in s, the SI units for acceleration are m/s ²   (meters per second  squared)

Recall that velocity is a vector —it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an Acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both.

ACCELERATION AS A VECTOR

Acceleration is a  vector in the same direction as the change in velocity, Δv. Since velocity is a vector , it can change either in magnitude or in direction. Acceleration  is therefore a change in either speed or direction, or both. Keep in mind that although acceleration  is in the direction of the change in velocity, it is not always in the direction of motion

. If acceleration  is in a direction opposite to the direction of motion , the object slows down.

Problem :

A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration ?

First we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity.

We can solve this problem by identifying Δv and Δt from the given information and then calculating the average acceleration directly from the equation

ặ=Δv / Δt=(V_f−V₀) / (T_f−T₀)

We know that

V₀=0, V_f =−15.0 m/s (the negative sign indicates direction toward the west), Δt=1.80 s

Since the horse is going from zero to −15.0 m/s, its change in velocity equals its final velocity: Δv=v_f=−15.0 m/s

Substituting the values , we  get

ặ  = (−15.0 m/s) / 1.80 s= −8.33 m/s²

Falling Objects

Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process.

Gravity

The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the same constant acceleration, independent of their mass. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones.

In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time . (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction  between objects—such as between clothes and a laundry chute or between a stone and a pool into which it is dropped—also opposes motion between them. For the ideal situations of these first few chapters, an object falling without air resistance or friction  is defined to be in free-fall . The forceof gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the acceleration due to gravity . The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so important that its magnitude is given its own symbol, g. It is constant at any given location on Earth and has the average value

g=9.80 m/s²

The direction of the acceleration due to gravity  is downward (towards the center of Earth). In fact, its direction defines what we call vertical. Note that whether the acceleration a in the kinematic equations has the Value +g or −g depends on how we define our coordinate system . If we define the upward direction as positive, then a=−g=−9.80 m/s² , and if we define the downward direction as positive, then a=g=9.80 m/s²

KINEMATIC EQUATIONS FOR OBJECTS IN FREE FALL :

V=V₀−gt

y=y₀+v₀ t−1/2(gt²)

v²=(v₀)²−2g(y−y₀)

Example problem :

A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s. The rock misses the edge of the cliff as it falls back to earth. Calculate the position  and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance .

Solution :

Draw a sketch

We are asked to determine the position y at various times. It is reasonable to take the initial position y₀ to be zero. This problem involves one-dimensional Motion  in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity  is downward, so a is negative. It is crucial that the initial velocity and the acceleration due to gravity  have opposite signs. Opposite signs indicate that the acceleration due to gravity  opposes the nitial motion and will slow and eventually reverse it.

Since we are asked for values of position  and velocity at three times, we will refer to these as y₁ and v₁ ; y₂ and v₂; and y ₃ and v₃

Solution for Position y₁ :

We know that y₀=0; v₀=13.0 m/s; a=−g=−9.80 m/s² and t=1.00 s.

We need to Identify the best equation to use. We will use 

y=y₀+v₀ t−1/2(gt²)

which is the value we want to find.

Substituting the known values and solve for y1

y1=0+(13.0 m/s)(1.00 s) – 1/ 2(−9.80 m/s2)(1.00 s)² =8.10 m

Solution for Velocity V₁ :

Given that 

y₀=0;  v₀=13.0 m/s;  a=−g=−9.80 m/s²

We also know from the solution above that y1=8.10 m

The best equation to use is v=v₀−gt

 Where (−g )= gravitational acceleration

Substituting the values , we get  

V₁=v₀ −gt=13.0 m/s−(9.80 m/s²)(1.00 s)=3.20 m/s

Solution for Remaining Times

The procedures for calculating the position and velocity att = 2.00 s and 3.00 s are the same as those above. The results are summarized in the Table below :

Time , t	Position , y	Velocity, v
1.00 s	8.10 m	3.20 m
2.00 s	6.40 m	−6.60 m
3.00 s	−5.10m	−16.4 m

Centripetal Acceleration :

We defined acceleration  as a change in velocity, either in its magnitude or in its direction, or both. When an object moves along a circular path, the direction of its velocity changes constantly, so there is always an associated acceleration, even if the speed of the object is constant. You experience this acceleration  yourself when you turn a corner in your car. What you notice is a sideways acceleration  because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration  will become. In this section we briefly examine the direction and magnitude of that acceleration.

The below image shows an object moving in a circular path at constant speed, called uniform circular motion . The direction of the instantaneous velocity  is shown at two points along the path. Acceleration  is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. We call the acceleration of an object moving in uniform circular motion  (resulting from a net external force) the centripetal acceleration (a_c ) – centripetal means “toward the center” or “center seeking.”

The directions of the velocity of an object at two different points are shown, and the change in velocity Δv is seen to point directly toward the center of curvature. (See small inset.) Because ac=Δv/Δt , the acceleration is also toward the center; ac is called centripetal acceleration. (For small time differences, Δθ is very small, and the arc length Δs is approximately equal to the chord length Δr)

The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? If we use the geometry shown in the above Figure along with some kinematics equations, we can obtain

ac=v² / r

which is the acceleration of an object in a circle of radius r at a speed v.

We see in the above Equation that centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that ac is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/h than at 50 km/h. A sharp corner has a small radius, so that a_c is greater for tighter turns .

The unit of centripetal acceleration is m/s²

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Introduction

Physics is a fascinating branch of science that seeks to understand the fundamental principles governing the behaviour of the universe. One important aspect of physics is the study of physical quantities and their dimensions. In this lesson, we will explore the concept of dimensions, delve into dimensional analysis, and discover its practical application

Dimensions of Physical Quantities

1.1 Definition:
Physical quantities are measurable properties or characteristics of objects or phenomena, such as length, time, mass, and Velocity

Dimensions represent the nature of these quantities and provide a framework for their measurement
1.2 Fundamental Dimensions :

There are seven fundamental dimensions in the International System of Units (SI): length (L), mass (M), time (T), electric current (I), temperature (Θ), amount of substance (N), and luminous intensity (J)

All other physical quantities can be derived from these fundamental dimensions

Quantity	Unit	Symbol	Definition
Length	Meter	m	The meter is the distance traveled by light in a vacuum in 1/299,792,458th of a second.
Mass	Kilogram	kg	The kilogram is defined as the mass of the International Prototype of the Kilogram, a platinum-iridium cylinder kept at the International Bureau of Weights and Measures.
Time	Second	s	The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.
Electric Current	Ampere	A	The ampere is the constant current that, if maintained in two parallel conductors of infinite length and negligible cross-sectional area, placed 1 meter apart in a vacuum, would produce a force between the conductors of exactly    2 x 10-7 newton per meter of length.
Temperature	Kelvin	K	The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
Amount of Substance	Mole	mol	The mole is the amount of a substance that contains as many elementary entities (such as atoms, molecules, ions, or particles) as there are atoms in exactly 0.012 kilograms of carbon-12.
Luminous Intensity	Candela	cd	The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

These fundamental units provide a standardized and universally accepted system of measurement for various physical quantities, allowing for consistency and accuracy in scientific calculations and communications.

Advantages of SI Units

The SI (International System of Units) unit system offers several advantages:

1. Universality: The SI unit system is widely recognized and accepted globally. It provides a consistent and unified framework for measurements across different countries and scientific disciplines. This universality promotes international collaboration, communication, and understanding in various fields of study.
2. Standardization: SI units have well-defined definitions and are based on fundamental constants of nature. This standardization ensures accuracy, reliability, and reproducibility in measurements. It allows for precise comparisons and consistent results in scientific experiments and observations.
3. Coherence: SI units are designed to be coherent, meaning they are interrelated in a logical and consistent manner. The coherent system of units simplifies calculations and conversions between different quantities, as they are based on consistent mathematical relationships.
4. Scalability: The SI unit system is scalable, meaning it provides prefixes that can be used to express measurements across a wide range of magnitudes. This scalability makes it convenient to work with both extremely large and small values without the need for excessive zeros or complex conversions.
5. Compatibility with Scientific Laws: SI units are compatible with the laws of physics and other scientific principles. They align with the fundamental theories and principles that govern the behavior of the physical world, allowing for the formulation and testing of scientific theories, laws, and equations.
6. Practicality: The SI unit system is designed to be practical and user-friendly. The units are intuitive and easily understandable, facilitating effective communication and comprehension of measurements and quantities among scientists, engineers, and the general public.
7. International Recognition: The SI unit system is officially recognized by international organizations such as the International Bureau of Weights and Measures (BIPM), ensuring consistent usage and adoption in scientific research, education, industry, and trade worldwide.
   These qualities make it an indispensable tool for precise and accurate measurements in scientific and technological advancements.

1.3 Derived Dimensions

Derived dimensions are obtained by combining the fundamental dimensions using multiplication, division, or exponentiation
For example, velocity has dimensions of [L][T]⁻¹, where [L] represents length and [T]⁻¹ represents the inverse of time
Here is a table of 22 derived SI units along with the basic units involved in their derivation:

Derived SI Unit	Quantity	Definition	Basic Units Involved
Newton (N)	Force	1 N = 1 kg * m/s2	kilogram (kg), meter (m), second (s)
Pascal (Pa)	Pressure	1 Pa = 1 N/m2	Newton (N), meter (m)
Joule (J)	Energy	1 J = 1 N * m = 1 kg * m2/s2	Newton (N), meter (m), second (s)
Watt (W)	Power	1 W = 1 J/s = 1 kg * m2/s3	Joule (J), second (s)
Coulomb (C)	Electric charge	1 C = 1 A * s	Ampere (A), second (s)
Volt (V)	Electric potential	1 V = 1 W/A = 1 kg * m2/(A * s3)	Watt (W), Ampere (A), second (s)
Ohm (Ω)	Electrical resistance	1 Ω = 1 V/A = 1 kg * m2/(A2 * s3)	Volt (V), Ampere (A), second (s)
Farad (F)	Capacitance	1 F = 1 C/V = 1 A*s/V =            1 kg-1 * m-2 * s4 * A2	Coulomb (C), Volt (V), Ampere (A), second (s), kilogram (kg), meter (m)
Tesla (T)	Magnetic flux density	1 T = 1 Wb/m^2 = 1 kg/(A * s2)	Weber (Wb), meter (m), Ampere (A)
Henry (H)	Inductance	1 H = 1 V * s/A = 1 kg * m2/(A2 * s2)	Volt (V), second (s), Ampere (A)
Hertz (Hz)	Frequency	1 Hz = 1/s	1/second (1/s)
Siemens (S)	Electrical conductance	1 S = 1 A/V = 1 s3 * A2/(kg * m2)	Ampere (A), Volt (V), second (s), kilogram (kg), meter (m)
Weber (Wb)	Magnetic flux	1 Wb = 1 V * s = 1 kg * m2/(A * s2)	Volt (V), second (s), kilogram (kg), meter (m), Ampere (A)
Lux (lx)	Illuminance	1 lx = 1 lm/m2 = 1 cd * sr/m2	Lumen (lm), meter (m), candela (cd), steradian (sr)
Becquerel (Bq)	Radioactivity	1 Bq = 1/s	1/second (1/s)
Gray (Gy)	Absorbed dose	1 Gy = 1 J/kg	Joule (J), kilogram (kg)
Sievert (Sv)	Equivalent dose	1 Sv = 1 J/kg	Joule (J), kilogram (kg)
Weber per square meter	Magnetic field strength	1 Wb/m2 = 1 T	Weber (Wb), meter (m)
Radian (rad)	Plane angle	1 rad = 1 m/m	Meter (m)
Steradian (sr)	Solid angle	1 sr = 1 m2/m2	Meter (m)
Celsius (°C)	Temperature	°C = (K – 273.15)	Kelvin (K)

Prefixes and Multiples of SI Units

Since the magnitude of the SI units vary over a wide range , multiples and sub multiples are used to explain the units more precisely

Here is a table of the standard prefixes for SI units, including both multiples and submultiples

Prefix	Symbol	Multiplication Factor
Yotta	Y	1024
Zetta	Z	1021
Exa	E	1018
Peta	P	1015
Tera	T	1012
Giga	G	109
Mega	M	106
Kilo	K	103
Hecto	H	102
Deca	da	101
–	–	100 (Base Unit)
Deci	D	10-1
Centi	C	10-2
Milli	M	10-3
Micro	µ	10-6
Nano	N	10-9
Pico	P	10-12
Femto	F	10-15
Atto	A	10-18
Zepto	z	10-21
Yocto	y	10-24

Additional Practical Units of the SI system

Here are some practical units of the SI system

Parsec: The parsec (pc) is a unit of length used in astronomy to measure vast distances to stars and galaxies. It is approximately equal to 3.09 x 10¹⁶ meters.

Light Year: The light-year (ly) is another unit of astronomical distance. It represents the distance light travels in one year, and it is about 9.46 x 10¹⁵ meters.

Astronomical Unit: The astronomical unit (AU) is a unit of length used in astronomy to describe distances within our solar system. It is approximately the mean distance from the Earth to the Sun and is about 1.5 x 10¹¹ meters.

Micron: The micron (μm) is a unit of length equal to one millionth of a meter. It is often used to measure small distances, such as the width of a human hair or microscopic objects.

Angstrom: The angstrom (Å) is a unit of length used to measure atomic and molecular scales. It is equal to 0.1 nanometers or 10^-10 meters.

Fermi: The fermi (fm) is a unit of length used in nuclear and particle physics to measure atomic and subatomic scales. It is equal to 10^-15 meters.

X-ray Unit: The X-ray unit (XU) is a unit used to measure the intensity of X-rays. It is a non-SI unit and is defined as the X-ray exposure that produces 2.58 x 10^-4 coulombs per kilogram of air.

Atomic Mass Unit (amu): The atomic mass unit (amu) is a unit used to express the mass of atoms and molecules on a scale relative to the mass of a carbon-12 atom. 1 amu is approximately 1.66 x 10^-27 kilograms.

These units are used in various scientific and practical contexts, especially in astronomy, physics, and other fields dealing with very large or very small distances and masses.

Methods of Measurement of Physical Quantities

The direct method and indirect method are two approaches used to measure physical quantities. Here’s a brief explanation of each method:

Direct Method: In the direct method of measurement, the physical quantity of interest is measured directly using appropriate instruments or techniques. The measurement is made by comparing the quantity being measured to a known standard or using a calibrated instrument. This method provides a straightforward and accurate measurement of the desired quantity. For example, measuring the length of an object using a ruler or measuring the temperature using a thermometer are examples of direct measurements.

Indirect Method: In the indirect method of measurement, the physical quantity of interest is derived or calculated by measuring other related quantities and using mathematical relationships or equations. This method involves measuring multiple quantities and utilizing mathematical models or formulas to determine the desired quantity indirectly. The indirect method is used when direct measurement of the quantity is not feasible or when it is more practical to measure other related quantities. For example, determining the velocity of an object by measuring its displacement and time, or calculating the area of an irregular shape by measuring its dimensions and applying appropriate formulas are examples of indirect measurements.

Both direct and indirect methods have their advantages and limitations. The choice of method depends on the nature of the physical quantity being measured, the available instruments or techniques, and the accuracy required for the measurement. In many cases, a combination of direct and indirect methods may be employed to obtain accurate and reliable measurements of complex physical quantities

Let us consider the measurement of the Length of an object . If the object is measureable by placing a scale , then its called the Direct Method of measurement . However this method is not feasible if we measure the height of a hill , for which we need to adapt indirect methods

Various methods of measurement of Length

Echo Method of Length Measurement :

The echo method of length measurement is a technique used to determine the distance to an object or surface by measuring the time it takes for a sound wave or pulse to travel to the object and back. This method is commonly employed in various applications, such as sonar systems, ultrasound imaging, and distance measurement devices.

Here’s how the echo method works:

A sound wave or pulse is generated and emitted towards the target object or surface.

The sound wave travels through the medium (such as air or water) until it reaches the object.

Upon reaching the object, the sound wave reflects or echoes back towards the source.

A receiver detects the returning sound wave or echo.

The time elapsed between the emission of the sound wave and the reception of the echo is measured.

The distance to the object is then calculated using the known speed of sound in the medium and the measured time.

Since the speed of sound in a given medium is relatively constant, knowing the time it takes for the sound wave to travel allows for the calculation of the distance based on the equation:

Distance = (Speed of Sound × Time) / 2

The division by 2 is necessary because the measured time accounts for the round trip of the sound wave.

The echo method can be used to measure distances accurately over a wide range, from short distances in medical ultrasound imaging to large distances in sonar systems for navigation or oceanographic research. It is a non-contact measurement method that is often preferred when direct physical contact with the object being measured is not possible or practical.

RADAR method of Length measurement :

The radar method of length measurement is a technique that utilizes radar (radio detection and ranging) to determine the distance to an object or surface. It is widely used in applications such as navigation, weather monitoring, and remote sensing.

Here’s how the radar method works:

A radar system emits short pulses of electromagnetic waves, typically radio waves or microwaves.

These waves travel at the speed of light and are directed towards the target object or surface.

Upon encountering the object, a portion of the electromagnetic waves is reflected back towards the radar system.

The radar system’s receiver detects the reflected waves, commonly referred to as radar echoes.

The time delay between the emission of the pulse and the reception of the echo is measured.

The distance to the object is then calculated using the known speed of light and the measured time delay.

The speed of light is approximately 299,792,458 meters per second (in a vacuum). By multiplying the speed of light by the measured time delay and dividing it by 2 (since the time accounts for the round trip), the radar system can accurately determine the distance to the target.

Radar systems can provide precise and reliable distance measurements over various ranges, from short distances to long distances. Additionally, radar can also provide additional information about the target, such as its velocity or size, by analyzing the Doppler shift or the strength of the radar echoes.

The radar method is widely used in applications such as air traffic control, maritime navigation, meteorology, and even in automotive systems like radar-based distance sensors for collision avoidance. It allows for non-contact measurement and provides valuable information in a wide range of scenarios.

SONAR method of Length measurement :

The sonar method of length measurement is a technique that uses sound waves to measure the distance between an object or surface and the source of the sound. Sonar (Sound Navigation And Ranging) is commonly used underwater to determine distances, map the ocean floor, and detect underwater objects. Here’s an overview of how the sonar method works:

Sound Wave Generation: A sonar system emits sound waves, usually in the form of short pulses, into the water or another medium. The sound waves can be produced by specialized sonar transducers

Sound Wave Propagation: The emitted sound waves travel through the medium, such as water, until they encounter an object or surface. The sound waves propagate through the medium at a known speed, which is typically slower than the speed of light.

Reflection or Echo: When the sound waves encounter an object or surface, a portion of the sound energy is reflected back towards the sonar system. This reflection is referred to as an echo.

Detection: The sonar system’s receiver detects the returning echoes. The receiver is designed to capture and analyze the reflected sound waves.

Time Measurement: The time delay between the emission of the sound wave and the reception of the echo is measured precisely. This time delay is often referred to as the “echo time” or “round-trip time.”

Distance Calculation: The distance between the sonar system and the object or surface is then calculated using the known speed of sound in the medium and the measured echo time. The distance is determined by multiplying half of the echo time by the speed of sound:

Distance = (Speed of Sound × Echo Time) / 2

Sonar is widely used in various applications, including underwater navigation, bathymetry (measurement of water depth), fish finding, underwater imaging, and locating underwater structures or objects. It allows for non-contact measurement over long distances underwater, providing valuable information about the environment beneath the surface

LASER method of Length measurement :

The laser method of length measurement utilizes laser technology to accurately measure distances with a high degree of precision. This method is widely employed in various fields, including engineering, surveying, manufacturing, and scientific research. Here’s an overview of how the laser method works:

Laser Emission: A laser emits a highly focused beam of light, which travels in a straight line.

Target Reflection: The laser beam is directed towards the target object or surface. When the laser beam encounters the target, a portion of the light is reflected back towards the laser source.

Detection: The reflected laser light is detected using a sensor or detector. The detector captures the returning light and converts it into an electrical signal.

Time Measurement: The time taken for the laser light to travel to the target and back is precisely measured using electronic timing equipment. This time measurement is usually accomplished using high-frequency electronic circuits or time-of-flight measurement techniques.

Speed of Light: The speed of light in air or a vacuum is a well-known constant (approximately 299,792,458 meters per second). By multiplying the speed of light by the measured time and dividing it by 2 (since the time accounts for the round trip), the distance to the target can be calculated.

Distance = (Speed of Light × Time) / 2

Data Processing: Multiple measurements are often taken to improve accuracy and account for any variations or errors. Advanced data processing techniques, including statistical analysis and error correction algorithms, may be employed to enhance the precision of the measurements.

The laser method of length measurement offers numerous advantages, including non-contact measurement, high accuracy, and rapid data acquisition. It is commonly used for tasks such as distance ranging, 3D scanning, alignment, surface profiling, and dimensional inspection. Additionally, the laser method can be applied in both short-range and long-range measurement scenarios, depending on the specific laser technology and setup used.

So we see that multiple indirect measurement methods are available for all practical purposes.

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