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

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.

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.

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=(Aω²=)0.12×157²=3.0×10³(ms^-2);  (watch for ECF from wrong period) 

(iv) at t=0.052sx=(-)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).

Candidates were asked to define SHM as applied to the situation in the question. Many failed to do this and wrote in general terms about SHM.

(i) This was well done.

(ii) Almost all candidates were able to identify a correct point for the maximum acceleration.

(iii) and (iv) Solutions for these were confused. Some attempted to use kinematic equations. Others mixed metres and centimetres in their answers. Other algebraic errors were present too (e.g. confusing 12² – 4 ² for (12 - 4)² ). This is an area that candidates could practice more.

(i) There were three marks for this question: for distinctions between longitudinal and transverse and for a clear description of the point of comparison. The latter was the mark most frequently lost. Many candidates have the vague idea that something about transverse is perpendicular and that the same parameter is parallel for longitudinal, but what “that something” is was frequently confused.

(ii) Candidates are now taking more care over the clear declaration of the frequency leading to the wavelength.