Define simple harmonic motion.

(i) On the axes below, sketch a graph to show how the acceleration of the mass varies with displacement from the equilibrium position B.

(ii) On your graph, label the points that correspond to the positions A, B and C.

(i) On the axes below, sketch a graph to show how the velocity of the mass varies with
time from the moment of release from A until the mass returns to A for the first time.

 (ii) On your graph, label the points that correspond to the positions A, B and C.

The period of oscillation is 0.20s and the distance from A to B is 0.040m. Determine the maximum speed of the mass.

A long spring is stretched so that it has a length of 10.0 m. Both ends are made to oscillate with simple harmonic motion so that transverse waves of equal amplitude but different frequency are generated.

Wave X, travelling from left to right, has wavelength 2.0 m, and wave Y, travelling from right to left, has wavelength 4.0 m. Both waves move along the spring at speed 10.0 m s^–1.

The diagram below shows the waves at an instant in time.

(i) State the principle of superposition as applied to waves.

(ii) By drawing on the diagram or otherwise, calculate the position at which the resultant wave will have maximum displacement 0.20 s later.

the force/acceleration is proportional to the displacement from the equilibrium position/centre;
the force/acceleration is directed towards the equilibrium position/centre / the force/acceleration is in the opposite direction to the displacement;

(i) straight line through the origin;
with negative gradient;

(ii) all three labels correct;

(i) positive sine graph;
drawn correctly for one period;

(ii) all three labels correct;
Accept either of the As and either of the Bs.
Accept either B if shown on the time axis in the correct position.

\(\omega  = \frac{{2\pi }}{T} = \frac{{2\pi }}{{0.20}} = 31.42 \approx 31{\rm{rad}}{{\rm{s}}^{ - 1}}\);
\({v_{\max }} = \omega {x_0} = 31.42 \times 0.040\);
\({v_{\max }} = 1.257 \approx 1.3{\rm{m}}{{\rm{s}}^{ - 1}}\);

(i) if two or more waves overlap/meet/pass through the same point;
the resultant displacement at any point is found by adding the displacements produced by each individual wave;

(ii) 0.20 s later, wave X will have crests at 5.0, 3.0 and 1.0 m, wave Y will have crests at 5.0 and 9.0 m / each wave will have moved forward by 2.0 m in 0.20 s / wave profiles for 0.20 s later drawn on diagram;

maximum displacement where two crests meet, i.e. at 5.0 m;