Physics 9702 · AS & A Level · Energy in simple harmonic motion

Energy in simple harmonic motion — practice question

A ball with mass $37\,\text{g}$ is trapped between fixed points A and B by two stretched helical springs, as illustrated in Fig. 2.1. It moves to and fro along AB in simple harmonic motion with frequency $3.5\,\text{Hz}$ and amplitude $2.8\,\text{cm}$.
(a)[2]

Demonstrate that the overall energy of the oscillations is $7.0\,\text{mJ}$.

(b)[3]

At two points during the ball's motion, its kinetic energy is the same as the potential energy in the springs. Work out the size of the displacement where this happens.

(c(i))[1]

Using the axes in Fig. 2.2 together with your answers to (a) and (b), draw a graph to display how the total energy of the system changes with displacement $x$ (label the line T).

(c(ii))[2]

On the same axes, draw a graph showing how the kinetic energy of the ball varies with displacement $x$ (label the line K).

(c(iii))[2]

On the same axes, draw a graph showing how the potential energy stored in the springs changes with displacement $x$ (label the line P).

(d)[1]

The arrangement shown in Fig. 2.1 is now turned through $90^{\circ}$ so that AB is vertical and the ball oscillates in a vertical plane. Suggest one form of energy, apart from those in (c), that has to be considered when drawing new graphs to show energy changes with displacement.

(c)[5]

Using the axes in Fig. 2.2 and your answers to (a) and (b), sketch graphs to show how displacement $x$ affects: (i) the total energy of the system (label this line T), (ii) the kinetic energy of the ball (label this line K), (iii) the potential energy stored in the springs (label this line P).

Worked solution & mark scheme

This 16-mark question has a full step-by-step worked solution and mark scheme. One marking point: Apply $E = \tfrac12 m\omega^2 a^2$

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