Oscillations & Waves

Q1. (3 marks)

• SHM: motion in which the restoring force (or acceleration) is directly proportional to the displacement from equilibrium and directed towards equilibrium. (1)
• $F = -kx$ (negative sign = restoring, towards equilibrium). (1)
• Conditions: (i) acceleration $\propto$ displacement; (ii) acceleration directed opposite to displacement (towards mean position). (1)

Q2. (4 marks) (a) Comparing with $x = A\cos(\omega t + \phi)$:

• $A = 0.05\ \text{m}$, $\omega = 4\pi\ \text{rad/s}$, $\phi = \pi/6$. (2)

(b) $T = \dfrac{2\pi}{\omega} = \dfrac{2\pi}{4\pi} = 0.5\ \text{s}$. (1) $f = \dfrac{1}{T} = 2\ \text{Hz}$. (1)

Q3. (4 marks) (a) $v = \omega\sqrt{A^2 - x^2} = 10\sqrt{0.08^2 - 0.04^2}$ $= 10\sqrt{0.0064 - 0.0016} = 10\sqrt{0.0048} = 10(0.0693) = 0.693\ \text{m/s}$. (2)

(b) $a_{max} = \omega^2 A = 10^2 \times 0.08 = 8\ \text{m/s}^2$. (2)

Q4. (5 marks)

• Restoring torque/force on bob displaced by angle $\theta$: tangential force $= -mg\sin\theta$. (1)
• Small-angle approximation: $\sin\theta \approx \theta$ (in radians), valid for small $\theta$. (1)
• Arc displacement $x = L\theta$, so $F = -mg\theta = -\dfrac{mg}{L}x$. (1)
• This is SHM with effective $k = mg/L$, so $\omega^2 = g/L$. (1)
• $T = \dfrac{2\pi}{\omega} = 2\pi\sqrt{\dfrac{L}{g}}$. (1)

Q5. (4 marks) (a) $\omega = \sqrt{\dfrac{k}{m}} = \sqrt{\dfrac{100}{0.25}} = \sqrt{400} = 20\ \text{rad/s}$. (1) $T = \dfrac{2\pi}{20} = 0.314\ \text{s}$. (1)

(b) $E = \tfrac12 k A^2 = \tfrac12 (100)(0.06)^2 = \tfrac12(100)(0.0036) = 0.18\ \text{J}$. (2)

Q6. (4 marks)

• Superposition principle: when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements. (2)
• Constructive: path difference $= n\lambda$ (integer multiple), waves in phase. (1)
• Destructive: path difference $= (n+\tfrac12)\lambda$, waves out of phase by $\pi$. (1)

Q7. (4 marks) Compare with $y = A\sin(kx - \omega t)$: $k = 6$, $\omega = 300$. (a) $\lambda = \dfrac{2\pi}{k} = \dfrac{2\pi}{6} = 1.047\ \text{m}$. (1.5) (b) $f = \dfrac{\omega}{2\pi} = \dfrac{300}{2\pi} = 47.75\ \text{Hz}$. (1.5) (c) $v = \dfrac{\omega}{k} = \dfrac{300}{6} = 50\ \text{m/s}$. (1)

Q8. (4 marks) (a) $f_{beat} = |260 - 256| = 4\ \text{Hz}$. (1) (b) The two waves superpose; because frequencies differ slightly, they periodically go in and out of phase, producing alternating maxima (loud) and minima (soft) in amplitude. The envelope oscillates at frequency $|f_1 - f_2|$, giving 4 beats per second. (3)

Q9. (4 marks) Source approaching stationary observer: $f' = f\dfrac{v}{v - v_s} = 500 \times \dfrac{340}{340 - 30}$ (2) $= 500 \times \dfrac{340}{310} = 500 \times 1.0968 = 548.4\ \text{Hz}$. (2)

Q10. (4 marks) (a) Q factor measures the sharpness/quality of an oscillator: $Q = 2\pi \times \dfrac{\text{energy stored}}{\text{energy lost per cycle}}$; high Q = low damping, slow energy loss. (2) (b) (2)

• Underdamped: oscillates with decaying amplitude.
• Critically damped: returns to equilibrium fastest without oscillating.
• Overdamped: returns to equilibrium slowly without oscillating.

[
{"claim":"Q3a speed at x=4cm is ~0.693 m/s","code":"import math\nv=10*sqrt(Rational(8,100)**2-Rational(4,100)**2)\nresult=abs(float(v)-0.6928)<0.01"},
{"claim":"Q5 total energy = 0.18 J","code":"E=Rational(1,2)*100*(Rational(6,100))**2\nresult=E==Rational(18,100)"},
{"claim":"Q7 wave speed = 50 m/s","code":"result=(300/6)==50"},
{"claim":"Q9 Doppler frequency ~548.4 Hz","code":"f=500*340/(340-30)\nresult=abs(float(f)-548.387)<0.01"}
]