IB Physics HL C.1 Simple harmonic motion questions

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Question 1

Paper 2Easy6 marks

A position sensor tracks a component undergoing simple harmonic motion. Its displacement xx from equilibrium is modelled by

x=Acos⁡ϕ,ϕ=ϕ0+2πtT,x=A\cos\phi,\qquad \phi=\phi_0+\frac{2\pi t}{T},

where the phase angle ϕ\phi is in radians. The amplitude is 0.060 m0.060\ \mathrm{m} the period is 0.80 s0.80\ \mathrm{s} and the initial phase angle is π/3\pi/3

(a)

State the phase angle at t=0t=0

[1]
(b)

Calculate the phase angle at t=0.20 st=0.20\ \mathrm{s} in radians.

[2]
(c)

Calculate the displacement of the component at t=0.20 st=0.20\ \mathrm{s}

[2]
(d)

State the increase in phase angle during one complete oscillation, in radians.

[1]

Question 2

Multiple choiceHard1 mark

An oscillator has a period of 2.0 s2.0\ \text{s} Four proposed graphs show its acceleration aa against displacement xx measured from equilibrium. Which trace is consistent with the oscillator undergoing simple harmonic motion?

-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5-6-5-4-3-2-10123456x / ma / m s-2PQRS
Show answer and explanation

Answer: A. For simple harmonic motion, the graph of aa against xx is a straight line through the origin with gradient −ω2-\omega^2 From T=2.0 sT=2.0\ \text{s} ω=2π/T=π rad s−1\omega=2\pi/T=\pi\ \text{rad s}^{-1} so the required gradient is −π2≈−9.87 s−2-\pi^2\approx-9.87\ \text{s}^{-2} Trace P has this negative gradient and passes through the origin.
B: uses a negative gradient through the origin, but its magnitude is about 3.14 s−23.14\ \text{s}^{-2} confusing ω\omega with ω2\omega^2
C: has the required gradient but does not pass through the origin, so acceleration is not proportional to displacement.
D: has a positive gradient, so acceleration is in the direction of displacement rather than towards equilibrium.
Remember: SHM requires acceleration to be proportional to displacement and opposite in direction.

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