Q no. 
Answer 
Commentary 
1 
E 
Match like terms with x = A cos (2pft) don’t worry about the “sin” it’s just a phase relationship x = a sin (wt) don’t panic about the “a” representing “A” then amplitude, A = a and angular speed, w = 2pf (i) at max amplitude acceleration is max … = w^{2}A (ii) momentum = mv = m (w Ö A^{2} – x^{2}) max v is when x=0 so mv = mwA (iii) KE = ½ mv^{2 } = ½ m (wA)^{2} (see above) (iv) w = 2pf rearranges to give f = w / 2p 
2 
B 
(i) object has greatest speed when x = 0. t_{2} is closer to x = 0 so FALSE (ii) greatest acceleration when x = A, t_{1} is closer to A (iii) at t_{1 }object is accelerating into the centre (it’s ALWAYS doing that except when x=0) (iv) no – it has the least acceleration when x = 0 so the acceleration is decreasing (even though it is still accelerating) 
3 
D 
a = w^{2}x if f = b/2p and w = 2pf then rearrange both to give w = b at extremity – x = A therefore acceleration = b^{2}A 
4 
C 
Spring system – T is unaltered (no g in the formula) so can only be B or C Pendulum – T depends upon g so must be C [also: (Ö 10 /Ö 40) = ½ ] 
5 
A 
divide the two expression for time period for a simple
pendulum: most terms cancel to give: Ö l / Ö (I + 1.8) = T / 2T rearrange to give l = 0.6m 
6 
C 
Beat frequency is the frequency of the consecutive instants when the pendulums are in phase. f_{beat} = f_{1} – f_{2} or 1/T_{beat } = 1/T_{1} – 1/T_{2} rearranges to give T_{beat} = 198 seconds The longer pendulum has a T = 2s In 198 seconds the longer pendulum will oscillate 99 times 
7 
B 
At x = 0 only energy is KE = ½ mv^{2 } This is equal to total energy via conservation At x=0 v reduces to v = 2pfA KE = ½ m(2pfA)^{2} 
8 
D 
Changing amplitude of driving force will change amplitude, energy and power but not frequency 
9 
C 
Conductor moving in a magnetic field will lead to eddy currents which will oppose the motion (Lenz’s law – remember magnets falling down the copper piping) 
10 

x = 0.05 sin 6t match like terms with x = A cos (2pft) don’t worry about sin or cos – it just a phase difference so: A = 0.05m w = 6 f = 6/2 p T = 1/f T = 2p / 6
max a is when x = A a = w^{2}A = 6^{2 }X 0.05 = 1.8ms^{1} Hookes law says that F is proportional to extension (ie displacement of mass) So: F = kx where k is the constant of proportionality F = ma rearranges to give a = (k/m)x negative because when you stretch a spring the force opposes the displacement – which is a definition of SHM Match like terms then w^{2} = (k/m) w = Ö(k/m) = 2pf rearranges to give f = (1/2p)Ö(k/m) time period T = 1/f = 2pÖ(m/k) forced vibrations: when an external periodic force is applied to a system that is free to oscillate resonance: when the forced frequency of vibration = natural frequency of the system – leads to a sharp peak in the amplitude of vibrations damping: any periodic loss of energy from a vibrating system at greater damping the resonance peak is broadened over a greater range of frequencies and the amplitude of the resonance peak is reduced ( see Breithaupt) 