Units and Measurements — Practice

Question 1

Find the dimensions of the universal gravitational constant G from F = G m1 m2 / r^2.

Accepted Answer

Rearrange for G: G = F r^2 / (m1 m2) = [MLT^-2][L^2]/[M^2] = [M^-1 L^3 T^-2]. Solving the defining equation for the unknown constant, then substituting each piece's own dimension, is the fastest route for any unfamiliar constant.

Question 2

The period of a pendulum T depends on length l, mass m of the bob, and g. Derive the relation using dimensional analysis, and state what the method cannot give you.

Accepted Answer

Assume T = k l^x g^y m^z. Then [T] = [L]^x [LT^-2]^y [M]^z, so matching powers: x + y = 0, -2y = 1, z = 0. Hence y = -1/2, x = 1/2, z = 0, giving T = k sqrt(l/g) with mass dropping out entirely. The method cannot give k; the full derivation for small oscillations gives k = 2*pi.

Question 3

A cube's side is measured as 7.203 m. Give its total surface area and volume to the correct number of significant figures.

Accepted Answer

The measured side has 4 significant figures, so both results round to 4. Surface area = 6(7.203)^2 = 311.299254 -> 311.3 m^2. Volume = (7.203)^3 = 373.714754 -> 373.7 m^3.

Question 4

How many significant figures are in 0.005060?

Accepted Answer

Four -- the digits 5, 0, 6, 0. The leading zeros are placeholders only; the final zero counts because it follows a decimal point.

Question 5

Check whether v^2 = u^2 + 2as is dimensionally correct.

Accepted Answer

LHS [L^2T^-2]; u^2 = [L^2T^-2]; 2as = [LT^-2][L] = [L^2T^-2]. All terms match, so the equation is dimensionally consistent -- though note this proves only that it is not wrong, not that it is right.

Question 6

Add 436.32 g, 227.2 g and 0.301 g, giving the answer to the correct precision. Why is the rule different from multiplication?

Accepted Answer

Raw sum = 663.821 g, but 227.2 g is known to only one decimal place, so the answer is 663.8 g. Addition and subtraction are limited by DECIMAL PLACES because it is the least precisely known decimal position that caps the sum; multiplication and division are limited by SIGNIFICANT FIGURES instead.

Question 7

Both K = (1/2)mv^2 and K = (3/16)mv^2 have the dimensions of energy. Explain why dimensional analysis cannot decide between them, and name two other things it cannot do.

Accepted Answer

Both reduce to [ML^2T^-2], and a dimensional check is blind to bare numerical factors, so both pass identically -- only the derivation of kinetic energy from the work-energy theorem settles it. The method also cannot handle relations involving sin, log or exponentials (whose arguments must already be dimensionless), and cannot distinguish two quantities sharing a dimension, such as torque and energy.

Question 8

Why was the kilogram redefined in 2018, and what is it now based on?

Accepted Answer

It was previously defined by a platinum-iridium cylinder kept near Paris -- a physical object that could be contaminated or damaged and so drift out of true. It is now defined by declaring the Planck constant to be exactly 6.62607015e-34 J s, so realising a kilogram becomes a reproducible measurement problem rather than a matter of matching one irreplaceable artefact.

Question 9

A rectangular sheet measures 16.2 cm by 10.1 cm on a metre scale. Give its area with the correct uncertainty.

Accepted Answer

Each reading carries +/- 0.1 cm, so l = 16.2 cm +/- 0.6% and b = 10.1 cm +/- 1%. For a product the percentage errors add: 0.6% + 1% = 1.6%. Area = 163.62 cm2, and 1.6% of that is 2.6 cm2. Round the uncertainty to one figure and match the value to it: area = 164 +/- 3 cm2.

Question 10

Two masses, 1.02 g and 9.89 g, are each measured to +/- 0.01 g on the same balance. Which has the larger relative error, and by how much?

Accepted Answer

Relative error = (uncertainty / value) x 100%. For 1.02 g: 0.01/1.02 x 100 = +/-1%. For 9.89 g: 0.01/9.89 x 100 = +/-0.1%. The smaller mass has ten times the relative error, even though the instrument and the absolute uncertainty are identical.

Practice — Units and Measurements

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