The correct answer is D because:

The centripetal force, F is given by the equation
- F = $\frac{m v^{2}}{r}$
- This is equal to the frictional force (since this is the force that keeps the girl staying in a circle)
The linear velocity, v is given by the equation
- v = ωr = $\frac{2 πr}{T}$
Substituting v into the centripetal force equation gives
- F = $\frac{m {(\frac{2 πr}{T})}^{2}}{r} = \frac{m \frac{4 π^{2} r^{2}}{T^{2}}}{r} = \frac{m 4 π^{2} r}{T^{2}}$
Rearranging for the time period, T
- T = $\sqrt{\frac{m 4 π^{2} r}{F}}$
Substituting in the values to calculate T gives
- T = $\sqrt{\frac{45 \times 4 \times π^{2} \times (130 \times 10^{- 2})}{630}}$ = 1.9146 = 1.9 s

A is incorrect as the 2π hasn’t been squared (to give 4π²)

B is incorrect as the r² on the numerator in $\frac{m \frac{4 π^{2} r^{2}}{T^{2}}}{r}$ has not be cancelled with the r on the denominator

C is incorrect as the square root has not been taken (which would give T² instead of T)

The easiest way to divide a fraction by a fraction is to rewrite it as follows:

$\frac{\frac{r^{2}}{T^{2}}}{r} = \frac{r^{2}}{T^{2}}$ ÷ r where r can also be written as $\frac{r}{1}$ (same for any whole number)

When dividing by a fraction (or whole number), multiply them together and flip the numerator and denominator of the second fraction

$\frac{r^{2}}{T^{2}}$ ÷ r = $\frac{r^{2}}{T^{2}} \times \frac{1}{r} = \frac{r^{2}}{T^{2} r} = \frac{r}{T^{2}}$

This makes it easier, since multiplying by two fractions is done by multiplying the numerators and multiplying the denominators.

Circular Motion (AQA A Level Physics)

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