1. 1
   An engineer tests electronic components on a vibration table. The flat, horizontal top of the table oscillates vertically with simple harmonic motion of amplitude 2.5 mm and frequency 50 Hz.

   (a)

   What is the maximum speed of the table top?

   [1 mark]

   • A0.13 m s⁻¹
   • B7.9 × 10² m s⁻¹
   • C2.5 × 10² m s⁻¹
   • D0.79 m s⁻¹

   (b)

   What is the magnitude of the maximum acceleration of the table top?

   [1 mark]

   • A6.3 m s⁻²
   • B2.5 × 10² m s⁻²
   • C0.79 m s⁻²
   • D2.5 × 10⁵ m s⁻²

   (c)

   A small component rests loosely on the table top. Deduce whether it stays in contact with the table throughout each oscillation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two identical mass–spring oscillators, P and Q, each oscillate with simple harmonic motion of period 0.60 s and amplitude 5.0 cm. The displacement of P, in metres, is given by x_P = 0.050 sin(ωt). Q reaches its maximum positive displacement 0.15 s before P does.

   (a)

   The displacement of Q is written as x_Q = 0.050 sin(ωt + ϕ). What is the phase angle ϕ?

   [1 mark]

   • Aπ/2 rad
   • Bπ/4 rad
   • Cπ rad
   • D0.15 rad

   (b)

   What is the speed of P when its displacement is 3.0 cm?

   [1 mark]

   • A0.21 m s⁻¹
   • B0.31 m s⁻¹
   • C0.42 m s⁻¹
   • D0.52 m s⁻¹

   (c)

   Determine the displacement of Q at t = 0.10 s.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A trolley of mass 0.20 kg is attached between two springs on a horizontal track and oscillates with simple harmonic motion. Friction is negligible. A motion sensor shows that 10 complete oscillations take 8.0 s and that the maximum speed of the trolley is 0.47 m s⁻¹.

   (a)

   Determine the amplitude and the total energy of the oscillation.

   [3 marks]

   (b)

   Determine the displacements at which the kinetic energy of the trolley equals its elastic potential energy, and explain why these are not at half the amplitude.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A grandfather clock is regulated by a simple pendulum of length 0.994 m with a bob of mass 1.5 kg. Where g = 9.81 m s⁻², the period is 2.00 s and the clock keeps perfect time; its mechanism counts each oscillation as exactly 2.00 s. The bob swings with an amplitude of 5.0 cm, measured along its arc. The clock is later moved to a mountain observatory where g = 9.77 m s⁻².

   (a)

   At the original location, time t = 0 is taken as an instant when the bob passes through its equilibrium position moving in the negative direction, and its displacement along the arc is modelled as x = x₀ sin(ωt + ϕ). Determine the phase angle ϕ, the maximum speed of the bob, its speed when it is 3.0 cm from equilibrium and its kinetic energy at that point.

   [6 marks]

   (b)

   Discuss whether the clock will keep accurate time at the observatory and how the clockmaker could correct it.

   [6 marks]

   Total for question 4: 12 marks

End of questions