JEE Physics Units and Dimensions Notes Set 02

Physical Quantities and Units

A physical quantity is a property that can be measured, and a measurement is stated as a number multiplied by a unit. Fundamental (base) quantities are independent of each other and the others are derived from them.
1. The seven SI base quantities and units are: length (metre, m), mass (kilogram, kg), time (second, s), electric current (ampere, A), temperature (kelvin, K), amount of substance (mole, mol) and luminous intensity (candela, cd).
2. Two supplementary quantities are plane angle (radian) and solid angle (steradian). Both are dimensionless.
3. Common systems: CGS (cm, g, s), MKS or SI (m, kg, s) and FPS (foot, pound, second).
4. Practical units of length: 1 light year = \( 9.46\times10^{15} \) m, 1 parsec = 3.26 light years = \( 3.08\times10^{16} \) m, 1 astronomical unit = \( 1.5\times10^{11} \) m, 1 angstrom = \( 10^{-10} \) m, 1 fermi = \( 10^{-15} \) m.
5. Other conversions: 1 N = \( 10^5 \) dyne, 1 J = \( 10^7 \) erg, 1 kg = 1000 g, 1 N/m\(^2\) = 10 dyne/cm\(^2\), 1 W h = 3600 J, 1 poiseuille = 10 poise.

Dimensions

The dimensions of a quantity are the powers to which the base quantities are raised to represent it. For mechanics the base dimensions are [M], [L] and [T].

Quantities such as angle, strain, refractive index, relative density and all trigonometric ratios are dimensionless. A dimensionless quantity may still have a unit (radian). A quantity that has no unit is always dimensionless.

Principle of Homogeneity and its Uses

Only quantities of the same dimensions can be added, subtracted or compared. The argument of a trigonometric, exponential or logarithmic function must be dimensionless. This gives three uses.
1. Checking an equation. A dimensionally wrong equation is certainly wrong. A dimensionally right one may still be wrong because numerical factors (1/2, 2π) have no dimensions.
2. Converting units. If \( n_1u_1 = n_2u_2 \) for a quantity with dimensions \( M^aL^bT^c \), then \( n_2 = n_1\left(\frac{M_1}{M_2}\right)^a\left(\frac{L_1}{L_2}\right)^b\left(\frac{T_1}{T_2}\right)^c \). A larger unit gives a smaller number.
3. Deriving relations. Assume the quantity is a product of powers of the variables it depends on, equate the powers of M, L and T and solve. Examples are the time period of a pendulum \( T \propto \sqrt{l/g} \) and the speed of waves in deep water. The method fails when more than three variables are involved without some being known, when a quantity depends on a sum of terms, and for constants that carry no dimensions.
Constants in an expression take dimensions that make each term homogeneous. For example in \( F = at + bt^2 \), a has \( MLT^{-3} \) and b has \( MLT^{-4} \).

Mathematics Used in Physics

1. Small angle approximation: for \( \theta \) in radian and small, \( \sin\theta \approx \tan\theta \approx \theta \) and \( \cos\theta \approx 1 - \frac{\theta^2}{2} \). The angle subtended by an object of size h at distance d is \( \theta = \frac hd \). Conversion: \( 1^\circ = \frac{\pi}{180} \) rad.
2. Binomial approximation: \( (1 + x)^n \approx 1 + nx \) when \( |x| \ll 1 \). Example: \( \sqrt{99} = 10(1 - 0.01)^{1/2} \approx 9.95 \).
3. Trigonometric values: sin 30° = 1/2, sin 37° = 3/5, sin 45° = \( \frac{1}{\sqrt2} \), sin 53° = 4/5, sin 60° = \( \frac{\sqrt3}{2} \), sin 15° = \( \frac{\sqrt3 - 1}{2\sqrt2} \).
4. Reduction formulas: \( \sin(90^\circ + \theta) = \cos\theta \), \( \sin(180^\circ - \theta) = \sin\theta \), \( \cos(180^\circ - \theta) = -\cos\theta \), and tan has period 180°, so tan 405° = tan 45° = 1.
5. A straight line \( y = mx + c \) has slope \( m = \tan\theta \), the angle with the positive x-axis. A line with equation \( 4x + 3y + 1 = 0 \) has slope \( -\frac43 \), x-intercept \( -\frac14 \) and y-intercept \( -\frac13 \).
6. Circle: circumference \( 2\pi r \), area \( \pi r^2 \). Sphere: surface \( 4\pi r^2 \), volume \( \frac43\pi r^3 \).

Significant Figures

1. All non-zero digits are significant. Zeros between non-zero digits are significant. Leading zeros are not. Trailing zeros after a decimal point are significant.
2. In multiplication and division, the result has as many significant figures as the factor with the fewest.
3. In addition and subtraction, the result has as many decimal places as the number with the fewest decimal places.
4. A vernier scale has least count = 1 main scale division minus 1 vernier division. If N vernier divisions equal N − 1 main divisions, the least count is (1 main division) / N.

Worked Example

The magnetic field B is found from \( F = qvB \). Then \( [B] = \frac{MLT^{-2}}{AT\cdot LT^{-1}} = MT^{-2}A^{-1} \). Since \( C = AT \), this is the same as \( MT^{-1}C^{-1} \).