Measurement

NUST NET (Engineering) · Physics · Measurement, vectors and equilibrium. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

Every physical quantity is a number multiplied by a unit. The seven SI base units are the metre, kilogram, second, ampere, kelvin, mole and candela. Every other unit, such as the newton, joule, pascal or coulomb, is built from these.

The dimensions of a quantity show how it depends on mass [M], length [L] and time [T]. Both sides of a correct equation must have the same dimensions, so you can test a formula, or find the dimensions of a constant by making it the subject.

No measurement is exact. The uncertainty in a result comes from the uncertainties in the readings, combined by two simple rules given below.

Rules to remember

Common mistake

Converting a squared or cubed unit only once. 1 cm2 = 10−4 m2 and 1 cm3 = 10−6 m3, so apply the prefix factor once for each power.

Worked example

The energy of a photon is E = hf, where f is frequency. What are the dimensions of Planck's constant h, and which mechanical quantity shares them?

  1. Make h the subject: h = E ÷ f.
  2. Energy has dimensions [ML2T−2] and frequency has [T−1].
  3. [h] = [ML2T−2] ÷ [T−1] = [ML2T−1].
  4. Angular momentum = mvr has [M][LT−1][L] = [ML2T−1], the same.

Answer: [ML2T−1], the same as angular momentum (unit J s).

Practice questions

Try each one, then open the answer.

1. The coulomb, the SI unit of electric charge, expressed in SI base units is

  1. A
    A s−1
  2. B
    A s
  3. C
    A−1 s
  4. D
    kg A s−2
Show answer

Answer: B. Charge = current × time, so 1 C = 1 A × 1 s = 1 A s. A s−1 would be current divided by time, which is not charge.

2. An acceleration of 1 cm ms−2 (one centimetre per millisecond squared) expressed in m s−2 is

  1. A
    10 m s−2
  2. B
    10−8 m s−2
  3. C
    106 m s−2
  4. D
    104 m s−2
Show answer

Answer: D. 1 cm = 10−2 m and 1 ms2 = (10−3 s)2 = 10−6 s2, so 1 cm ms−2 = 10−2 ÷ 10−6 = 104 m s−2. Forgetting to square the millisecond gives 10 m s−2.

3. Using F = Gm1m2/r2, the dimensions of the gravitational constant G are

  1. A
    [M−1L3T−2]
  2. B
    [ML3T−2]
  3. C
    [M−2L3T−2]
  4. D
    [M−1L2T−2]
Show answer

Answer: A. G = Fr2/(m1m2) = [MLT−2][L2] ÷ [M2] = [M−1L3T−2]. Taking Fr2 alone, without dividing by the two masses, gives [ML3T−2].

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