Units, dimensions and errors for NEET physics

"Physics and measurement" is the first unit of the NEET physics syllabus, and one of the quickest to finish. Its questions are usually short: a dimensional formula, a count of significant figures, a percentage error, or an instrument reading. They are also easy to get wrong in a hurry, because each has one small rule that students half-remember.

The unit pays off beyond its own questions. A dimensional check takes ten seconds and can knock out two wrong options in almost any chapter. This guide covers the rules properly, then works through the question types you will meet.

SI units: base and derived

The SI system has seven base quantities. Every other unit is built from these.

Plane angle (radian) and solid angle (steradian) are dimensionless. Derived units come from formulas: the newton is kg m s⁻², the joule is N m, the watt is J s⁻¹.

Dimensional formulas worth knowing

Work each one out from a defining formula once, so you can rebuild it in the hall instead of recalling it.

Three uses of dimensional analysis

1. Checking an equation

By the principle of homogeneity, every term added or equated must have the same dimensions. In s = ut + ½at², each term is [L]: [u][t] = [L T⁻¹][T] and [a][t²] = [L T⁻²][T²]. The ½ is a pure number and does not affect the check.

The same idea gives a quick rule: the argument of sin, cos, log or eˣ must be dimensionless. In y = A sin(ωt − kx), both ωt and kx have no dimensions, so ω is [T⁻¹] and k is [L⁻¹].

2. Converting units

Since a physical quantity does not change, n₁u₁ = n₂u₂. To convert 1 joule to ergs, write [M L² T⁻²] and substitute 1 kg = 10³ g and 1 m = 10² cm: 1 J = 10³ × (10²)² erg = 10⁷ erg.

3. Deriving a relation

Suppose the period T of a simple pendulum depends on the bob's mass m, the length l and g. Write T = k mᵃ lᵇ gᶜ. Then [T] = Mᵃ Lᵇ (L T⁻²)ᶜ. Matching powers: M gives a = 0; T gives −2c = 1, so c = −½; L gives b + c = 0, so b = ½. So T = k√(l/g). Dimensions cannot give k (it is 2π).

Limits of the method. It cannot find dimensionless constants. It cannot handle relations containing trigonometric, exponential or logarithmic functions. It cannot tell apart quantities with the same dimensions (work and torque). And with only M, L and T, it can solve for at most three unknown powers.

Significant figures

The rules, as NCERT states them:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant (2.05 has 3).
• Leading zeros before the first non-zero digit are not (0.0034 has 2).
• Trailing zeros after a decimal point are significant (3.500 has 4).
• Trailing zeros in a number without a decimal point are not significant (123 m written as 12300 cm still has 3). Scientific notation removes the doubt: 1.23 × 10⁴ cm.
• Changing units does not change the number of significant figures.

In calculations:

• Multiplication and division: keep as many significant figures as the least precise value. 4.237 g ÷ 2.51 cm³ = 1.688… → 1.69 g cm⁻³ (three figures, from 2.51).
• Addition and subtraction: keep as many decimal places as the value with the fewest. 436.32 + 227.2 + 0.301 = 663.821 → 663.8 (one decimal place, from 227.2).

Errors and how they combine

For repeated readings, the mean absolute error is the average of the absolute deviations from the mean. Relative error = Δa/a, and percentage error = (Δa/a) × 100.

Least count: vernier and screw gauge

This links to the experimental skills unit.

• Vernier callipers: least count = 1 MSD − 1 VSD. If 10 vernier divisions equal 9 main-scale divisions of 1 mm, LC = 1 − 0.9 = 0.1 mm.
• Screw gauge: least count = pitch ÷ number of circular-scale divisions. Pitch 0.5 mm with 50 divisions gives LC = 0.01 mm.
• Zero error: corrected reading = observed reading − zero error (with its sign). A positive zero error is subtracted; a negative one is effectively added.

Worked numericals

Example 1: accuracy of g from a pendulum

An aspirant measures L = 50.0 cm with a scale of 1 mm accuracy, and the time for 100 oscillations as 140 s with a watch of 1 s resolution. What is the percentage error in g?

• g = 4π²L/T², so Δg/g = ΔL/L + 2(ΔT/T).
• ΔL/L = 0.1/50.0 = 0.2%.
• Timing 100 oscillations divides the error by 100 as well as the time, so ΔT/T = 1/140 ≈ 0.71%.
• Δg/g = 0.2 + 2 × 0.71 ≈ 1.6%.

Timing many oscillations is why the percentage error in T stays small.

Example 2: dimensions of constants

In F = a√x + bt², F is force, x is distance and t is time. Find the dimensions of a/b.

• [a] = [F]/[x¹ᐟ²] (x to the power ½) = [M L T⁻²]/[L¹ᐟ²] = [M L¹ᐟ² T⁻²].
• [b] = [F]/[t²] = [M L T⁻⁴].
• [a/b] = [M L¹ᐟ² T⁻²] ÷ [M L T⁻⁴] = [L⁻¹ᐟ² T²].

Example 3: screw gauge with zero error

A screw gauge has pitch 0.5 mm and 50 circular divisions. With the jaws closed, the zero of the circular scale is 3 divisions below the reference line (a positive zero error). Measuring a wire, the linear scale reads 3.5 mm and the circular scale 27.

• LC = 0.5/50 = 0.01 mm. Zero error = +3 × 0.01 = +0.03 mm.
• Observed reading = 3.5 + 27 × 0.01 = 3.77 mm.
• Corrected = 3.77 − 0.03 = 3.74 mm.

Practice MCQs

1. The dimensions of the coefficient of viscosity are: (a) [M L⁻¹ T⁻¹] (b) [M L T⁻¹] (c) [M L⁻² T⁻²] (d) [M L⁻¹ T⁻²]
2. Which pair does not have the same dimensions? (a) work and torque (b) Planck's constant and angular momentum (c) stress and Young's modulus (d) impulse and force
3. The number of significant figures in 0.02030 is: (a) 2 (b) 3 (c) 4 (d) 5
4. V = (100 ± 5) V and I = (10 ± 0.2) A. The percentage error in R = V/I is: (a) 3% (b) 5% (c) 7% (d) 10%
5. If the units of force and length are each made four times larger, the unit of energy becomes: (a) unchanged (b) 4 times (c) 8 times (d) 16 times
6. In a vernier callipers, 20 VSD coincide with 19 MSD of 1 mm each. The least count is: (a) 0.01 mm (b) 0.05 mm (c) 0.1 mm (d) 0.5 mm
7. The error in measuring the side of a cube is 1%. The error in its volume is: (a) 1% (b) 2% (c) 3% (d) 6%

Answers

1. (a) η = F/(6πrv) = [M L T⁻²]/([L][L T⁻¹]) = [M L⁻¹ T⁻¹].
2. (d) Impulse is [M L T⁻¹]; force is [M L T⁻²].
3. (c) Leading zeros do not count; 2, 0, 3 and the trailing 0 after the decimal do.
4. (c) 5/100 = 5% and 0.2/10 = 2%; relative errors add to 7%.
5. (d) Energy = force × length, so 4 × 4 = 16.
6. (b) LC = 1 MSD − 1 VSD = 1 − 19/20 = 0.05 mm.
7. (c) V = a³, so 3 × 1% = 3%.

What to do next

• Derive every dimensional formula in the table above from its defining equation, without looking.
• Solve ten error-propagation questions and ten significant-figure questions from previous NEET papers.
• Revise vernier and screw-gauge readings from your practical file, including zero error.
• Use a dimensional check on at least one option in every numerical you solve this week.

Next in mechanics: kinematics for NEET. For the wider plan, see a physics plan for NEET.