JEE Physics Units and Dimensions Notes Set 01

Physical Quantities and Units

A physical quantity is anything that can be measured, such as length, mass, time, force or temperature. A measurement is a number together with a unit. Fundamental (base) quantities are independent of each other and the rest are derived from them. The SI system has seven base quantities: length (metre), mass (kilogram), time (second), electric current (ampere), temperature (kelvin), amount of substance (mole) and luminous intensity (candela). Two supplementary quantities, plane angle (radian) and solid angle (steradian), are dimensionless.

Older systems use only three base units: CGS (centimetre, gram, second), MKS (metre, kilogram, second) and FPS (foot, pound, second). Useful conversions: 1 micron = 10⁻⁶ m, 1 angstrom = 10⁻¹⁰ m, 1 fermi = 10⁻¹⁵ m, 1 light year = 9.46 × 10¹⁵ m, 1 parsec = 3.08 × 10¹⁶ m (a unit of distance), 1 newton = 10⁵ dyne, 1 joule = 10⁷ erg and 1 eV = 1.6 × 10⁻¹⁹ J. When the unit changes, the numerical value changes inversely: n₁u₁ = n₂u₂.

Dimensions

The dimensions of a physical quantity are the powers to which the base quantities are raised in its formula. For example, velocity has dimensions [M⁰L¹T⁻¹], force [M¹L¹T⁻²], work and energy [M¹L²T⁻²], pressure [M¹L⁻¹T⁻²], Planck's constant [M¹L²T⁻¹], gravitational constant [M⁻¹L³T⁻²] and surface tension [M¹T⁻²]. Angle, strain, refractive index and relative density are dimensionless.

The principle of homogeneity states that every term of a correct equation has the same dimensions. Only quantities of the same dimensions can be added or subtracted, and the argument of sin, log or exponential functions must be dimensionless. Dimensional analysis is used to (1) check a formula, (2) convert units from one system to another and (3) derive a relation among quantities. It cannot give numerical constants and cannot be used with trigonometric, logarithmic or exponential relations or when a quantity depends on the sum of terms.

Errors in Measurement

Systematic errors act in one direction and include instrumental error (for example zero error), error of method and personal error. Random errors vary unpredictably and are reduced by repeating measurements and averaging. Absolute error is |a(mean) − a(measured)|; the mean absolute error is the average of these; relative (fractional) error is the mean absolute error divided by the mean value; percentage error is the relative error multiplied by 100.

Combination of errors: for a sum or difference Z = A ± B the maximum absolute error is ΔZ = ΔA + ΔB. For a product or quotient, the fractional errors add. For Z = Aᵖ Bᵠ / Cʳ the maximum fractional error is ΔZ/Z = p ΔA/A + q ΔB/B + r ΔC/C. For example, in g = 4π²ℓ/T², the fractional error is Δℓ/ℓ + 2ΔT/T.

Significant Figures and Measuring Instruments

All non-zero digits are significant. Zeros between non-zero digits and trailing zeros after the decimal point are significant, while leading zeros are not. In multiplication or division the result has as many significant figures as the least precise factor; in addition or subtraction the result keeps as many decimal places as the least precise number. The order of magnitude of a quantity is the power of ten closest to it.

Vernier callipers: least count = value of 1 main scale division ÷ number of vernier divisions. If N vernier divisions equal (N − 1) main scale divisions, the least count is 1 MSD/N. Screw gauge: least count = pitch ÷ number of circular scale divisions. Final reading = main scale reading + (coincident division × least count) − zero error, where the zero error is signed, so a negative zero error is added back.

Solved Examples

Example 1. The density of a material is 4 g/cm³ in the CGS system. Find its value in a system where the unit of length is 10 cm and the unit of mass is 100 g.
Answer:
1 g = 0.01 new units of mass and 1 cm = 0.1 new unit of length.
Density = 4 × 0.01/(0.1)³ = 0.04/0.001 = 40 in the new units.

Example 2. The resistance R = V/I is calculated from V = (100 ± 5) V and I = (10 ± 0.2) A. Find the percentage error in R.
Answer:
Percentage error in V = 5%; percentage error in I = 0.2/10 × 100 = 2%.
For a quotient the errors add, so the error in R is 7%.

Example 3. Use dimensions to show that the time period of a simple pendulum depends on \( \sqrt{\ell/g} \).
Answer:
Assume \( T = k\,\ell^a g^b \). Then \( [T] = [L]^a[LT^{-2}]^b \).
Comparing powers of T: 1 = −2b so b = −1/2. Comparing powers of L: 0 = a + b so a = 1/2.
So \( T \propto \sqrt{\ell/g} \); the constant 2π cannot be found by dimensions.