 # SHM multi choice questions from old exam collection

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 … = w2A

(ii)  momentum = mv = m (w Ö A2 – x2)  max v is when x=0 so mv = mwA

(iii)  KE = ½ mv2   = 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.  t2 is closer to x = 0 so FALSE

(ii)  greatest acceleration when x = A, t1 is closer to A

(iii)  at t1 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  = -w2x  if  f = b/2p and  w = 2pf  then rearrange both to give w = b

at extremity – x = A  therefore acceleration = -b2A

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 1040) = 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.

fbeat = f1 – f2

or 1/Tbeat  = 1/T1 – 1/T2

rearranges to give Tbeat = 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 mv2

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  = w2A = 62  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 w2 = (k/m)

w = Ö(k/m) = 2pf  rearranges to give f = (1/2p)Ö(k/m)

time period T = 1/f = 2(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)