Q3. A trolley of mass 0.80 kg rests on a horizontal surface attached to two identical stretched springs, as shown in the diagram. Each spring has a spring constant of 30 Nm^–1, can be assumed to obey Hooke's law, and to remain in tension as the trolley moves.

(a)

(i) The trolley is displaced to the left by 60 mm and then released. Show that the magnitude of the resultant force on it at the moment of release is 3.6 N.

For one spring

change in force ΔF = kΔL

ΔF = 30 × 60 × 10^–3

ΔF = 1.8 N

The on spring pulls in one direction - the other in the opposite. The force from the streched spring pulls to a greater extent: [F + ΔF]; the force from the compressed spring pulls to a lesser extent [F – ΔF]

The overall force is the difference between these two forces.

Resultant force = [F + ΔF] – [F – ΔF]) = 2ΔF

2ΔF = 2 x 1.8 = 3.6 N QED

(2 marks)

(ii) Calculate the acceleration of the trolley at the moment of release and state its direction.

F = ma

a = F/m = ^3.6/_0.8 = 4.5 ms^–2

to the right

(2 marks)

(b)

(i) The oscillating trolley performs simple harmonic motion. State the two conditions which have to be satisfied to show that a body performs simple harmonic motion.

This gets asked for in most SHM questions - make sure you know it! The examiner wants you to define the equation for SHM in words - the proportionality and the meaning of negative sign.

Acceleration is proportional to the displacement from the equilibrium position

but acceleration is in the opposite direction to the displacement [or acceleration is always directed towards a fixed point/equilibrium position]

(2 marks)

(ii) The frequency f of oscillation of the trolley is given by

where

m = mass of the trolley

k = spring constant of one spring.

Calculate the period of oscillation of the trolley, stating an appropriate unit.

f = = 1.38 Hz (needs to be 3sf as it is an intermediate stage)

period T = ¹/_f = =¹/_1.38 = 0.72 s (for the unit)

(3 marks)

(c) Copper ions in a crystal lattice vibrate in a similar way to the trolley, because the inter-atomic forces act in a similar way to the forces exerted by the springs. The diagram below shows how this model of a vibrating ion can be represented.

(i) The spring constant of each inter-atomic 'spring' is about 200 Nm^–1. The mass of the copper ion is 1.0 × 10^–25 kg. Show that the frequency of vibration of the copper ion is about 10¹³ Hz.

Here you have a similar situation to the trolley so you have to use the given equation for the 'push - pull' dual-spring system given in part b.

= 1.0 x 10^-13 Hz QED

(1 mark)

(ii) If the amplitude of vibration of the copper ion is 10^–11m, estimate its maximum speed.

v_max = 2πfA

v_max= 2π × 10¹³ × 10^–11

v_max= 628 (3sf) need this to carry forward to next part

v_max= 630 ms^–1 (2sf)

(1 mark)

(iii) Estimate the maximum kinetic energy of the copper ion.

max E_K = ½ mv_max ²

max E_K= ½ × 1.0 × 10^–25 × 628²

max E_K= 2.0 × 10^–20 J

(1 mark)

(Total 12 marks)