A student is investigating forced vertical oscillations in springs - AQA - A-Level Physics - Question 7 - 2017 - Paper 1

To find the spring constant k, we can use the formula for maximum kinetic energy:

$KE = \frac{1}{2} k A^2$

Given that $KE = 54 \text{ mJ} = 54 \times 10^{-3} \text{ J}$ and $A = 2.5 \times 10^{-2} \text{ m}$, we can rearrange to find k:

$54 \times 10^{-3} = \frac{1}{2} k (2.5 \times 10^{-2})^2$

Solving for k:

$k = \frac{2 \times 54 \times 10^{-3}}{(2.5 \times 10^{-2})^2} = \frac{0.108}{6.25 \times 10^{-4}} = 172.8 \text{ N m}^{-1}$

For the mass m, using the formula for the angular frequency:

$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

Rearranging to find m:

$m = \frac{k}{(2\pi f)^2}$

Substituting k and f = 2.0 Hz:

$m = \frac{172.8}{(2\pi \cdot 2)^2} = \frac{172.8}{16\pi^2} \approx 0.33 \text{ kg}$