Physics is a science of measurement. Every physical quantity, whether it is the length of a pencil, the speed of a car, or the mass of the Sun, is expressed in terms of a number and a unit. A unit is a standard amount of a physical quantity chosen for convenience, and any measurement compares the quantity with this chosen standard. Without well-defined units, scientific communication would be impossible - one scientist's "large" could be another scientist's "small".
To bring order to measurement, scientists around the world agreed upon an international system of units known as the SI (Systeme International d'Unites) system. The SI system is based on seven fundamental or base quantities, from which all other physical quantities can be derived. This chapter introduces these base units, the concept of dimensions, the rules of significant figures, and the methods of estimating and handling errors in measurement.
The ability to express any physical law in terms of dimensions and to check the consistency of equations is one of the most powerful tools in physics. This chapter therefore lays the mathematical foundation on which all of mechanics and indeed all of physics is built.
The SI system defines seven base quantities and their units. These are the kilogram (kg) for mass, the metre (m) for length, the second (s) for time, the ampere (A) for electric current, the kelvin (K) for temperature, the mole (mol) for amount of substance, and the candela (cd) for luminous intensity.
Along with the base units, the SI system uses prefixes to express very large or very small quantities. For example, 1 kilometre = 10^3 m, 1 millimetre = 10^-3 m, 1 microsecond = 10^-6 s, and 1 nanometre = 10^-9 m. Common prefixes include kilo (k, 10^3), mega (M, 10^6), giga (G, 10^9), centi (c, 10^-2), milli (m, 10^-3), micro (mu, 10^-6), nano (n, 10^-9), and pico (p, 10^-12).
Some useful distance units used in astronomy are the astronomical unit (AU), the light year (ly), and the parsec (parsec). One AU is the average distance between the Earth and the Sun, about 1.496 x 10^11 m. One light year is the distance light travels in one year, about 9.46 x 10^15 m. One parsec is about 3.08 x 10^16 m, and 1 parsec = 3.26 light years.
The dimensions of a physical quantity are the powers to which the fundamental units of mass, length, and time are raised to express that quantity. Dimensions are written using square brackets. For example, the dimensions of velocity are written as [M^0 L T^-1] or simply [LT^-1], meaning length divided by time.
The dimension of force is found from Newton's second law, F = ma. Since acceleration has dimensions [LT^-2], force has dimensions:
[F] = [M][LT^-2] = [M L T^-2]
Similarly, the dimensions of energy (work) are the same as those of force times distance, giving [M L^2 T^-2].
Dimensional analysis is used for three main purposes. First, it checks the correctness or consistency of a physical equation - every term in a correct equation must have the same dimensions. Second, it can derive relationships between physical quantities. Third, it can convert a quantity from one system of units to another. However, dimensional analysis cannot determine dimensionless constants and cannot be applied to trigonometric, exponential, or logarithmic functions.
The number of significant figures in a measured quantity is the number of digits in which we have reasonable confidence. They convey the precision of the measurement. The rules for counting significant figures are: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are not significant; and trailing zeros are significant only if the number contains a decimal point.
For example, 0.0045 has two significant figures (4 and 5), 45.00 has four, and 4.5000 has five. When we round off, if the digit to be dropped is less than 5, we leave the preceding digit unchanged; if it is 5 or greater, we increase the preceding digit by one.
In calculations, the result should not have more significant figures than the least precise number used. In addition and subtraction, the result should be rounded to the least number of decimal places. In multiplication and division, the result should contain the same number of significant figures as the quantity having the fewest significant figures.
No measurement is perfectly accurate. The difference between the measured value and the true value is called error. Errors are classified as systematic errors (due to faulty instruments or incorrect technique) and random errors (due to unpredictable fluctuations). Systematic errors shift all readings in the same direction, while random errors scatter readings around the true value.
For a quantity measured n times with values a1, a2, ..., an, the mean value is:
a_mean = (a1 + a2 + ... + an) / n
The absolute error of each measurement is the magnitude of the difference between the measured value and the mean value. The mean absolute error is the average of all absolute errors. The relative error is the ratio of the mean absolute error to the mean value, and the percentage error is the relative error multiplied by 100.
If two quantities with errors are combined, errors add. For addition or subtraction, the absolute errors add; for multiplication, division, or powers, the relative errors add.
Precision and accuracy are different concepts. Precision refers to the closeness of the measurements to each other, while accuracy refers to the closeness of the measured value to the true value. A measurement can be precise without being accurate. For example, a faulty but consistent instrument can give readings that are very close to each other (high precision) but far from the true value (low accuracy).
In an experiment, precision is indicated by the number of significant figures in the result. The accuracy of a result is determined by the extent of systematic error present. Good measurement requires both high precision and high accuracy.
| Base Quantity | SI Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
| Luminous intensity | candela | cd |
| Physical Quantity | Dimensional Formula | SI Unit |
|---|---|---|
| Velocity | [L T^-1] | m/s |
| Acceleration | [L T^-2] | m/s^2 |
| Force | [M L T^-2] | kg m/s^2 (N) |
| Work and Energy | [M L^2 T^-2] | kg m^2/s^2 (J) |
| Pressure | [M L^-1 T^-2] | kg/(m s^2) (Pa) |
| Power | [M L^2 T^-3] | kg m^2/s^3 (W) |
In this chapter we learned the foundation of all measurements in physics. We studied the seven SI base units, from the metre to the candela, and understood how all other quantities are derived from them. The concept of dimensions allows us to check the correctness of equations and derive relationships, and we saw that the dimensions of force are [M L T^-2] and of energy are [M L^2 T^-2]. We mastered the rules of significant figures and the estimation of errors, distinguishing between systematic and random errors and between precision and accuracy. Together these tools let us express and evaluate measurements quantitatively, preparing us for the study of motion in a straight line.