Paper 4 Oscillations and SHM Answers

Q1 Measuring spring oscillator period

1. For a spring-mass oscillator,

$$T=2\pi \sqrt{\frac{m}{k}},$$

so

$$T^2=\frac{4{\pi }^2}{k}m.$$

Thus $T^2$ against $m$ is linear. [2] 2. Correct $T^2$ values, from $0.0225$ to $0.0853{\mathrm{s}}^2$. [1] 3. Axes and units [1]; suitable scale [1]; accurate plots [2]; best-fit straight line [1]. [5]

4. The fitted gradient is approximately $0.250{\mathrm{s}}^2\mathrm{k}{\mathrm{g}}^{-1}$. Since $G=4{\pi }^2/k$,

$$k=\frac{4{\pi }^2}{G}\approx 158\mathrm{N}{\mathrm{m}}^{-1}.$$

Accept follow-through from the candidate’s line. [3] 5. Writing $T^2=G(m+m_0)=Gm+G{m}_0$, the intercept is $G{m}_0$. With an intercept of about $0.0100{\mathrm{s}}^2$,

$$m_0=\frac{0.0100}{0.250}=0.040\mathrm{kg}.$$

[2] 6. Time many oscillations and divide by the number; keep amplitude small; use the same reference point and direction; repeat and average; avoid parallax if using video or a fiducial marker. Any two. [2]

Q2 Resonance experiment

1. It is the driving frequency at which the steady-state amplitude is maximum. [1]
2. Immediately after changing frequency, transient motion may still be present. Waiting for steady state ensures the measured amplitude corresponds to that driving frequency. [2]
3. The peak becomes lower and broader, and may shift slightly to lower frequency. [1]
4. Keep amplitudes within safe limits, secure the oscillator, avoid resonance amplitudes that damage the apparatus, or keep fingers clear of moving parts. [1]