The correct answer is C because:

The equation for the time period T of a simple pendulum is T = 2π $\sqrt{\frac{l}{g}}$
The equation for the gravitational field strength g is g = $\frac{G M}{r^{2}}$
It’s useful to write this equation in terms of the density ρ = $\frac{M}{V}$ using the equation for the volume of a sphere (V = $\frac{4}{3}$ πr³):
- g = $\frac{G M}{r^{2}} = \frac{G (ρ V)}{r^{2}} = \frac{G ρ (\frac{4}{3} {πr}^{3})}{r^{2}} = \frac{4}{3} \frac{G ρ {πr}^{3}}{r^{2}} = \frac{4}{3} πGρr$
Therefore, the gravitational field strength on another planet can be written as g’, where:
- g’ = $\frac{4}{3}$ πGρ’r’
We are told that the new density ρ’ = 2ρ and the new radius r’ = $\frac{1}{3}$ r
Substituting this into the equation for g’ is as follows:
- g' = $\frac{4}{3}$ πGρ’r’
- g' = $\frac{4}{3} πG (2 ρ) (\frac{1}{3} r)$ = $\frac{2}{3}$ × ( $\frac{4}{3}$ πGρr) = g
Determine the multiple of the gravitational field strength on Earth, g:
- g' = $\frac{2}{3}$ × ( $\frac{4}{3}$ πGρr) = g
Therefore, the new planet’s gravitational field strength g’, in terms of Earth’s gravitational field strength g is given by g’ = $\frac{2}{3}$ g
Since the time period of the pendulum on the new planet can be written as T’ = 2π $\sqrt{\frac{l}{g'}}$ we substitute g’ = $\frac{2}{3}$ g in to get:
- T’ = 2π $\sqrt{\frac{l}{g'}}$
- T’ = 2π $\sqrt{\frac{l}{(\frac{2}{3} g)}}$ = 2π $\sqrt{\frac{3 l}{2 g}}$ = $\sqrt{\frac{3}{2}}$ × 2π $\sqrt{\frac{l}{g}}$ = $\sqrt{\frac{3}{2}}$ T

Questions like these, in which you have to consider factor changes to quantities within an equation (e.g. radius doubles, mass quadruples, etc) are super common. The method outlined above is a really powerful technique that you should practise applying to these types of questions: writing the same equation but using a ‘dash’ (or other symbol) to differentiate between ‘old’ radius and ‘new’ radius, for example, is crucial. We used an apostrophe above, so that (in terms of new density ρ’ and old density ρ) we could write ρ’ = 2ρ. Be really careful when substituting and manipulating these equations.

Simple Harmonic Motion (AQA A Level Physics)

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