AQA A-Level Physics: Further Mechanics (Circular Motion and SHM)

A trolley of mass 0.40 kg is attached between two springs on a frictionless horizontal track. When the trolley is displaced sideways and released, it oscillates with simple harmonic motion about a central equilibrium position.

Describe and explain the motion of the trolley as it completes one full oscillation. In your answer you should:

• state the defining condition for simple harmonic motion and relate it to the forces acting on the trolley;
• explain how the kinetic and potential energy interchange during a cycle while the total energy stays constant;
• identify where in the cycle the speed and the acceleration each reach a maximum.

(6 marks)

A 0.250 kg mass hangs from a spring of spring constant 40.0 N m⁻¹. The mass is pulled down 0.060 m below its equilibrium position and released from rest, so that it performs simple harmonic motion.

Quantity	Symbol	Value
Mass	$m$m	0.250 kg
Spring constant	$k$k	40.0 N m⁻¹
Amplitude	$A$A	0.060 m

(a) Calculate the angular frequency $\omega$ω and the period $T$T of the oscillation. (2 marks)

(b) Calculate the maximum speed and the maximum acceleration of the mass. (2 marks)

(c) Calculate the displacement of the mass 0.10 s after it is released. (2 marks)

A glider on a horizontal air track oscillates with simple harmonic motion between two springs. Its motion is described as follows:

• The glider starts at the centre of its path and first returns to the centre, moving in the same direction, 0.80 s later.
• The farthest the glider gets from the centre in either direction is 0.12 m.
• A data-logger records that the glider's maximum kinetic energy during the motion is 0.96 J.

(a) State the amplitude of the oscillation and use the description to find its period, and hence calculate the angular frequency $\omega$ω. (3 marks)

(b) State the total energy of the oscillating glider, giving a reason. (1 mark)

(c) Calculate the speed of the glider when its displacement from the centre is 0.060 m. (1 mark)

A car of mass 1200 kg drives around a flat (unbanked) circular bend of radius 35 m. The road is dry and the maximum frictional force between the tyres and the road is 0.60 times the car's weight (coefficient of friction 0.60). Take $g = 9.81 \ \text{m s}^{-2}$g=9.81 m s−2.

(a) Explain why there is a maximum speed at which the car can go round the bend, and calculate this maximum speed. (3 marks)

(b) A second car of mass 1800 kg drives round the same bend. State, with a reason, how its maximum cornering speed compares with that of the 1200 kg car. (2 marks)

A laboratory centrifuge spins a small sample tube in a horizontal circle of radius 0.080 m. The rotor turns at 3000 revolutions per minute. The sample has a mass of 0.015 kg.

(a) Calculate the angular speed $\omega$ω of the sample. (2 marks)

(b) Calculate the centripetal acceleration of the sample. (1 mark)

(c) Calculate the centripetal force acting on the sample. (1 mark)

A particle moves with simple harmonic motion.

(a) State the condition that defines simple harmonic motion. (1 mark)

(b) Describe the phase relationship between the particle's velocity and its displacement, and between its acceleration and its displacement. (2 marks)