Units and Measurements

Chapter overview

Physics studies measurable quantities. A physical quantity is expressed as:

$$\text{Physical quantity}=\text{numerical value}\times\text{unit}$$

Example:

$$l=5\ \text{m}$$

Here, $5$ is the numerical value and $\text{m}$ is the unit.

1. Fundamental and derived quantities

Fundamental quantities are independent quantities. Derived quantities are obtained from fundamental quantities.

2. Derived units

3. Dimensions

Dimensions represent the powers of fundamental quantities in a physical quantity.

Example:

$$F=ma$$

Dimensional formula:

$$[F]=[M][LT^{-2}]=[MLT^{-2}]$$

4. Dimensional analysis

Principle of homogeneity

Only physical quantities with the same dimensions can be added or subtracted.

If:

$$A+B=C$$

then:

$$[A]=[B]=[C]$$

Example:

$$s=ut+\frac{1}{2}at^2$$

Checking dimensions:

$$[ut]=[LT^{-1}][T]=[L]$$ $$[at^2]=[LT^{-2}][T^2]=[L]$$

So the equation is dimensionally correct.

5. Uses of dimensional analysis

To check correctness of equations

Example:

$$T=2\pi\sqrt{\frac{l}{g}}$$

Dimension of RHS:

$$\left[\sqrt{\frac{l}{g}}\right] = \sqrt{\frac{[L]}{[LT^{-2}]}} = \sqrt{[T^2]} = [T]$$

So the formula is dimensionally correct.

To derive relations

Suppose time period of a simple pendulum depends on length $l$ and acceleration due to gravity $g$.

$$T\propto l^a g^b$$

So:

$$[T]=[L]^a[LT^{-2}]^b$$ $$[T]=[L^{a+b}T^{-2b}]$$

Equating powers:

$$a+b=0,\qquad -2b=1$$

Therefore:

$$b=-\frac{1}{2},\qquad a=\frac{1}{2}$$

Hence:

$$T\propto\sqrt{\frac{l}{g}}$$

6. Limitations of dimensional analysis

• It cannot determine dimensionless constants like $2$, $\pi$, or $\frac12$.
• It cannot distinguish between scalar and vector quantities.
• It cannot derive equations involving trigonometric, exponential, or logarithmic functions.
• It cannot verify equations where dimensions are correct but physics is wrong.

7. Significant figures

Significant figures are meaningful digits in a measured quantity.

Examples:

Rules:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.
• Trailing zeros after decimal are significant.

8. Accuracy and precision

Accuracy means closeness to the true value.

Precision means closeness among repeated measurements.

9. Errors in measurement

Measured value:

$$a_1,a_2,a_3,\ldots,a_n$$

Mean value:

$$\bar{a}=\frac{a_1+a_2+\cdots+a_n}{n}$$

Absolute error:

$$\Delta a_i=|a_i-\bar{a}|$$

Mean absolute error:

$$\Delta a_{\text{mean}} = \frac{\Delta a_1+\Delta a_2+\cdots+\Delta a_n}{n}$$

Relative error:

$$\frac{\Delta a}{a}$$

Percentage error:

$$\frac{\Delta a}{a}\times100$$

10. Combination of errors

Addition and subtraction

If:

$$Z=A+B$$

or:

$$Z=A-B$$

then:

$$\Delta Z=\Delta A+\Delta B$$

Multiplication and division

If:

$$Z=AB$$

or:

$$Z=\frac{A}{B}$$

then:

$$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$$

Powers

If:

$$Z=A^n$$

then:

$$\frac{\Delta Z}{Z} = n\frac{\Delta A}{A}$$

General rule:

$$Z=A^pB^qC^r$$

then:

$$\frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}$$

11. Vernier calipers and screw gauge

Least count of vernier calipers:

$$LC=\frac{1\ \text{MSD}}{\text{number of divisions on vernier scale}}$$

Measurement:

$$\text{Reading} = \text{main scale reading} + \text{vernier coincidence}\times LC$$

Screw gauge least count:

$$LC=\frac{\text{pitch}}{\text{number of circular scale divisions}}$$

Screw gauge reading:

$$\text{Reading} = \text{pitch scale reading} + \text{circular scale reading}\times LC$$

12. Solved examples

Example 1

Check the dimensions of:

$$v=u+at$$

Solution:

$$[v]=[LT^{-1}]$$ $$[u]=[LT^{-1}]$$ $$[at]=[LT^{-2}][T]=[LT^{-1}]$$

All terms have the same dimension, so the equation is dimensionally correct.

Example 2

If:

$$l=2.0\pm0.1\ \text{m}$$

and:

$$b=3.0\pm0.2\ \text{m}$$

then area:

$$A=lb$$

Relative error:

$$\frac{\Delta A}{A} = \frac{0.1}{2.0} + \frac{0.2}{3.0}$$

13. Common mistakes

• Confusing units and dimensions.
• Writing SI symbols in plural form, like $\text{mts}$ instead of $\text{m}$.
• Ignoring significant zeros after decimal.
• Adding percentage errors for addition/subtraction problems.
• Forgetting that dimensional analysis cannot determine constants.

14. Quick revision