Units and Measurements — Practice

Question 1

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

Accepted Answer

G = F r^2 / (m1 m2) = [MLT^-2][L^2]/[M^2] = [M^-1 L^3 T^-2].

Question 2

The period of a pendulum T depends on length L and g. Derive the relation using dimensional analysis.

Accepted Answer

Assume T = k L^a g^b. [T] = [L]^a [LT^-2]^b. Matching: a + b = 0, -2b = 1, so b = -1/2, a = 1/2. Thus T = k sqrt(L/g).

Question 3

Five time readings are 2.02, 2.04, 1.98, 2.00, 2.01 s. Find the mean absolute error.

Accepted Answer

Mean = 2.01 s. Absolute errors: 0.01, 0.03, 0.03, 0.01, 0.00. Mean absolute error = 0.08/5 = 0.016 s.

Question 4

How many significant figures are in 0.005060?

Accepted Answer

Four -- the digits 5, 0, 6, 0 (leading zeros are not significant; the trailing zero after the decimal is).

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 correct.

Practice — Units and Measurements