Paper 2 Oscillations and SHM Answers

Q1 SHM quantities

1. $A=0.060\mathrm{m}$. [1]
2. $\omega =8.0\mathrm{rad}{\mathrm{s}}^{-1}$. [1]
3. $T=2\pi /\omega =2\pi /8.0=0.785\mathrm{s}$. [2]
4. $v_{\max }=\omega A=8.0(0.060)=0.480\mathrm{m}{\mathrm{s}}^{-1}$. [2]
5. $a_{\max }={\omega }^{2}A=8.0^2(0.060)=3.84\mathrm{m}{\mathrm{s}}^{-2}$. [2]

Q2 Testing for SHM

1. Plot acceleration $a$ against displacement $x$. [1]
2. The graph should be a straight line through the origin with negative gradient. [2]
3. Since $a=-{\omega }^{2}x$, gradient $=-{\omega }^2$, so $\omega =\sqrt{-\text{gradient}}$. [2]
4. SHM requires acceleration towards equilibrium. By Newton’s second law, this requires a resultant restoring force towards equilibrium proportional to displacement. [2]

Q3 Damping and resonance

1. Amplitude decreases with time. [1]
2. Mechanical energy is transferred to internal energy of the oscillator/surroundings, and sometimes sound, due to dissipative forces. [2]
3. Resonance occurs when the steady-state amplitude is maximum as driving frequency is varied, usually near the natural frequency for light damping. [2]
4. Greater damping lowers the maximum amplitude and makes the resonance peak broader; the resonant frequency may shift slightly lower. [2]