### Question

This question is about simple harmonic motion (SHM).

The graph shows the variation with time \(t\) of the acceleration \(a\) of an object X undergoing simple harmonic motion (SHM).

a

Define *simple harmonic motion (SHM)*.[2]

X has a mass of 0.28 kg. Calculate the maximum force acting on X.[1]

Determine the maximum displacement of X. Give your answer to an appropriate number of significant figures.[4]

A second object Y oscillates with the same frequency as X but with a phase difference of \(\frac{\pi }{4}\). Sketch, using the graph opposite, how the acceleration of object Y varies with \(t\).[2]

**▶️Answer/Explanation**

### Markscheme

a

force/acceleration proportional to the displacement/distance from a (fixed/equilibrium) point/mean position;

directed towards this (equilibrium) point / in opposite direction to displacement/ distance;

*Allow algebra only if symbols are fully explained.*

0.73 (N); *(allow answer in range of 0.71 to 0.75 (N)) *

use of \({a_0} = – {\omega ^2}{x_0}\);

\(T = 7.9{\text{ (s)}}\) ** or** \(\omega = 0.795\)

**\(\frac{\pi }{4}{\text{ (rad}}\,{{\text{s}}^{ – 1}})\); }**

*or**(allow answers in the range of T = 7.8 to 8.0 (s)*

**\(\omega \)**

*or*

*=**0.785 to 0.805 (rad s*

^{–1}*))*

\({x_0} = 4.1(1){\text{ (m)}}\); *(allow answers in the range of 4.0 to 4.25 (m))*

two significant figures in final answer whatever the value;

*Award **[4] **for a bald correct answer.*

shape correct, constant amplitude for new curve, minimum of 10 s shown; } *(there must be some consistent **lead or lag and no change in T)*

lead/lag of 1 s (to within half a square by eye);

Simple harmonic motion and forced oscillations

The graph shows the variation with time of the displacement of an object undergoing simple harmonic motion.

a

(i) State the amplitude of the oscillation.

(ii) Calculate the frequency of the oscillation.[3]

(i) Determine the maximum speed of the object.

(ii) Determine the acceleration of the object at 140 ms.[4]

The graph below shows how the displacement of the object varies with time. Sketch on the same axes a line indicating how the kinetic energy of the object varies with time.

You should ignore the actual values of the kinetic energy.

[3]

**▶️Answer/Explanation**

## Markscheme

a

(i) 32 (mm);

(ii) period = 160 (ms);

frequency = 6.2/6.3 (Hz);

*Allow ECF for incorrect period. *

(i) *ω*=2π×6.25;*v*(=39.3×32×10^{-3})=1.3(ms^{-1});* (allow ECF from (a))*

**or **

tangent drawn to graph at a point of zero displacement;

gradient calculated between 1.2 and 1.4;

(ii) displacement = 23–26 (mm);

35–40 (ms^{-2});

*23 mm found by calculating displacement*

double frequency;

always positive and constant amplitude;

correct phase *ie* cosine squared;

*Ignore amplitude value.*

*A minimum of one complete, original oscillation needed to award [3].*

This question is in **two** parts. **Part 1** is about simple harmonic motion (SHM) and waves. **Part 2** is about wind power and the greenhouse effect.

**Part 1** Simple harmonic motion (SHM) and waves

A gas is contained in a horizontal cylinder by a freely moving piston P. Initially P is at rest at the equilibrium position E.

a

The piston P is displaced a small distance *A* from E and released. As a result, P executes simple harmonic motion (SHM).

Define *simple harmonic motion* as applied to P.

The graph shows how the displacement *x* of the piston P in (a) from equilibrium varies with time* t.*

(i) State the value of the displacement *A* as defined in (a).

(ii) On the graph identify, using the letter M, a point where the magnitude of the acceleration of P is a maximum.

(iii) Determine, using data from the graph and your answer to (b)(i), the magnitude of the maximum acceleration of P.

(iv) The mass of P is 0.32 kg. Determine the kinetic energy of P at* t*=0.052 s.[7]

The oscillations of P initially set up a longitudinal wave in the gas.

(i) Describe, with reference to the transfer of energy, what is meant by a longitudinal wave.

(ii) The speed of the wave in the gas is 340 m s^{–1}. Calculate the wavelength of the wave in the gas.[4]

**▶️Answer/Explanation**

## Markscheme

a.

the acceleration of piston/P is proportional to its displacement from equilibrium;

and directed towards equilibrium;*There must be a clear indication what is accelerating otherwise award *** [1 max]**.

(i) 12(cm); (*accept* –12)

(ii) any maximum or minimum of the graph;

(iii) period= 0.04 (s);* (allow clear substitution of this value)*

\(\omega = \left( {\frac{{2\pi }}{T} = } \right)\frac{{2 \times 3.14}}{{0.04}} = 157\left( {{\rm{rad }}{{\rm{s}}^{ – 1}}} \right)\)

maximum acceleration=(*Aω*^{2}=)0.12×157^{2}=3.0×10^{3}(ms^{-2}); *(watch for ECF from wrong period) *

(iv) at *t*=0.052s*x*=(-)4(±1)cm;

\({\rm{KE = }}\left( {\frac{1}{2}m{\omega ^2}\left[ {{A^2} – {x^2}} \right] = } \right)0.5 \times 0.32 \times {157^2}\left[ {{{0.12}^2} – {{0.04}^2}} \right] = 50\left( { \pm 7} \right)\left( {\rm{J}} \right)\);

*Watch for incorrect use of cm. Allow ECF from calculations in (b)(iii).Do not retrospectively credit a mark for ω to (b)(iii) if it was not gained there on original marking.Allow use of sin ωt to obtain v. Award *

**[2]**for a bald correct answer.

(i) the direction of the oscillations/vibrations/movements of the particles (in the medium/gas);

for a longitudinal wave are parallel to the direction of the propagation of the energy of the wave;

(ii) \(f = \left( {\frac{1}{T} = } \right)\frac{1}{{0.04}} = 25\left( {{\rm{Hz}}} \right)\);

\(\lambda = \left( {\frac{v}{f} = } \right)\frac{{340}}{{25}} = 14\left( {\rm{m}} \right)\);

*Award [1 max] if frequency is not clearly stated. Allow ECF from calculations in (b)(iii).*