Units, Dimensions and Measurement — Practice

Question 1

The dimensional formula of force is:
(a) MLT⁻² (b) ML²T⁻² (c) MLT⁻¹ (d) ML⁻¹T⁻²

Accepted Answer

(a) MLT⁻².

Force = mass × acceleration = M × LT⁻² = MLT⁻².

Question 2

Which quantity has the same dimensions as work?
(a) Torque (b) Power (c) Force (d) Momentum

Accepted Answer

(a) Torque.

Work = energy = ML²T⁻². Torque = force × distance = MLT⁻² × L = ML²T⁻², the same dimensions (though a different physical quantity).

Question 3

The dimensional formula of pressure is:
(a) ML⁻¹T⁻² (b) MLT⁻² (c) ML²T⁻² (d) ML⁻¹T⁻¹

Accepted Answer

(a) ML⁻¹T⁻².

Pressure = force/area = MLT⁻² / L² = ML⁻¹T⁻².

Question 4

The number of significant figures in 0.00340 is:
(a) 3 (b) 5 (c) 2 (d) 6

Accepted Answer

(a) 3.

Leading zeros are not significant; the significant digits are 3, 4 and the trailing 0 (which counts because it follows the decimal), giving 3 significant figures.

Question 5

A quantity Z = A²B/C. The percentage errors in A, B and C are 1%, 2% and 3% respectively. The maximum percentage error in Z is:
(a) 7% (b) 6% (c) 5% (d) 9%

Accepted Answer

(a) 7%.

ΔZ/Z = 2(ΔA/A) + (ΔB/B) + (ΔC/C) = 2(1%) + 2% + 3% = 7%. The square on A doubles its 1% contribution.

Question 6

The dimensional formula of Planck's constant h (from E = hν) is:
(a) ML²T⁻¹ (b) ML²T⁻² (c) MLT⁻¹ (d) ML²T⁻³

Accepted Answer

(a) ML²T⁻¹.

h = E/ν = (ML²T⁻²) / (T⁻¹) = ML²T⁻¹, the same dimensions as angular momentum.

Question 7

Which equation is dimensionally consistent? (s = distance, u = initial velocity, a = acceleration, t = time)
(a) s = ut + ½at² (b) s = ut + ½at (c) s = u + at² (d) s = ut² + ½at

Accepted Answer

(a) s = ut + ½at².

Each term must have dimension L. [ut] = LT⁻¹·T = L; [½at²] = LT⁻²·T² = L. Both match [s] = L, so the equation is dimensionally consistent (the ½ is dimensionless).

Question 8

One nanometre expressed in metres is:
(a) 10⁻⁹ m (b) 10⁻⁶ m (c) 10⁻³ m (d) 10⁻¹² m

Accepted Answer

(a) 10⁻⁹ m.

The prefix nano means 10⁻⁹, so 1 nm = 10⁻⁹ m (micro = 10⁻⁶, pico = 10⁻¹²).

Question 9

The dimensional formula of the universal gravitational constant G (from F = Gm₁m₂/r²) is:
(a) M⁻¹L³T⁻² (b) M⁻¹L²T⁻² (c) MLT⁻² (d) M⁻¹L³T⁻¹

Accepted Answer

(a) M⁻¹L³T⁻².

G = Fr²/(m₁m₂) = (MLT⁻²)(L²)/(M·M) = ML³T⁻²/M² = M⁻¹L³T⁻².

Question 10

The product 2.0 × 3.00, expressed to the correct number of significant figures, is:
(a) 6.0 (b) 6.00 (c) 6 (d) 6.000

Accepted Answer

(a) 6.0.

The result of a multiplication keeps the fewest significant figures of the inputs. 2.0 has 2 sig figs and 3.00 has 3, so the answer 6.00 is rounded to 2 sig figs: 6.0.

Question 11

The dimensional formula of power is:
(a) ML²T⁻³ (b) ML²T⁻² (c) MLT⁻² (d) ML²T⁻¹

Accepted Answer

(a) ML²T⁻³.

Power = energy/time = ML²T⁻² / T = ML²T⁻³.

Question 12

A length is measured as 2.0 m with an absolute error of 0.1 m. The percentage error is:
(a) 5% (b) 10% (c) 0.5% (d) 2%

Accepted Answer

(a) 5%.

Percentage error = (absolute error / measured value) × 100 = (0.1/2.0) × 100 = 5%.

Question 13

The dimensional formula of frequency is:
(a) T⁻¹ (b) T (c) LT⁻¹ (d) MT⁻¹

Accepted Answer

(a) T⁻¹.

Frequency = 1/time period, so its dimension is T⁻¹ (SI unit hertz, s⁻¹).

Question 14

The number of significant figures in the measurement 0.00340 m is:
(a) 3 (b) 5 (c) 2 (d) 6

Accepted Answer

(a) 3.

Leading zeros are not significant, but the trailing zero after the decimal is: the significant figures are 3, 4 and 0 — three in total.

Question 15

A quantity Z = A²B/C. If the percentage errors in A, B and C are 1%, 2% and 3% respectively, the maximum percentage error in Z is:
(a) 7% (b) 6% (c) 4% (d) 12%

Accepted Answer

(a) 7%.

For a product/quotient, fractional errors add with the powers as weights: %error(Z) = 2×%A + %B + %C = 2(1) + 2 + 3 = 7%.

Practice — Units, Dimensions and Measurement

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