AL-Essay-SHM / p.1
1.
(88-IIA-6)
(a) By considering the uniform motion of a point in a circular path, demonstrate the meaning
of simple harmonic motion (S.H.M.), giving a relation between the acceleration and
displacement from the equilibrium position for a particle undergoing S.H.M.
(3 marks)
(b) Briefly explain how it is possible to set a small object into S.H.M. using a spiral spring.
(3 marks)
(c) Atoms of a diatomic molecule (each of mass m) are able to oscillate towards and away
from each other in a similar manner to two small objects connected by a spiral spring.
Assume that the potential energy of such a system, for an atomic spacing x, is
U  (a / x)  b /( 2 x 2 ) , where a and b are constants.
(i) Obtain an expression for the corresponding force F between the two atoms and
determine the equilibrium separation in terms of a and b. Sketch out the variation of
F with x, explaining the shape and significant features of your graph.
(ii) Show that for small oscillations about the equilibrium location of the atoms (you
may consider one atom to remain stationary and the effective mass of the other atom
to be m/2), S.H.M. occurs and obtain expressions for
(I) the force constant (force/displacement) and
(II) the oscillation period.
(Hint : use binomial expansion for terms in the force equation.)
(10 marks)

2.

(90-IIA-1)
(a) Describe simple harmonic motion (s.h.m.).
(2 marks)
(b) A simple pendulum consisting of a weight suspended vertically by a string, of length l ,
attached to a fixed point is set in motion in a vertical plane, the amplitude of oscillations
being small. Show that the motion is simple harmonic and write down expressions for the
displacement, velocity and acceleration of the weight after a time t. Sketch the variations
of potential and kinetic energies with time.
(6 marks)
(c) Describe an experiment to verify that such a pendulum undergoes s.h.m.
(6 marks)
(d) A student decides to use the oscillation of such a pendulum to obtain a value for the freefall acceleration due to gravity. Without describing this experiment, critically discuss
TWO possible sources of error in your measurement.
(2 marks)
3.
(92-IIA-6)
(a) Define simple harmonic motion (s.h.m.) and give one example of motion approximating
to s.h.m. which may be observed during our daily lives.
(2 marks)
(b) (i) A long spiral spring of force constant k hangs vertically from a fixed support with a
weight of mass m attached to its bottom end. If the weight is pulled downwards and
then released show that the subsequent motion is s.h.m., with the displacement from
the equilibrium position at any time t given by x  a cos 0t , where a is a constant
and  0 the natural angular frequency of oscillation.
AL-Essay-SHM / p.2
(ii) If the weight now moves in a viscous liquid, there will be an additional retardation
dx
dx
force acting of b , where
is the instantaneous velocity. Write down the new
dt
dt
equation of motion. Assume a solution, x  Ae t cos t and by substitution derive
an expression for  and show that the new angular frequency of oscillation,
1
2 2
  ( 0   ) . (Hint: consider particular times t  0,
2

.)
2
(10 marks)
(c) Now consider the connection of a charged capacitor of capacitance C across a coil of
inductance L and resistance R. Show that a similar differential equation holds good for
the charge Q on the capacitor, in place of the displacement x in (b)(ii), and hence by
analogy determine the corresponding expression for  ,  and  0 .
(4 marks)
4.
(93-IIA-6)
(a) An ideal spring of force constant k is mounted horizontally with one end fixed and the
other end attached to a block of mass m (as shown). The block is set to oscillate with
amplitude A on a level, frictionless surface.
(i)
Sketch a graph of elastic potential energy U against x for the spring-mass system,
where x is the distance of the block from the fixed end. Mark on the graph the
position x c of the central point of the oscillatory motion.
(ii) On the same graph in (i), use a dotted line to sketch the graph of kinetic energy of
the system against x. Briefly explain your graph by using energy-based arguments.
(4 marks)
(b) In (a), the spring is assumed to have negligible mass. However, no spring is completely
massless. To find the effect of the spring’s mass, consider a spring of mass M and force
constant k. When the spring is stretched or compressed by an amount e, the elastic
potential energy is 1 ke 2 .
2
(i) Suppose at a certain instant, the speed of the block is v and the length of the spring is
L. For each spring element, its speed is proportional to its distance l from the fixed
end. Show that the kinetic energy of the spring is Mv2/6. (Assume uniform mass
distribution of the spring.)
(Hint : find the mass and the speed of each spring element of length dl.)
(ii) Find the extension of the spring in terms of x and write down the expression for the
total energy of such an oscillating spring-mass system. Take the time derivation of
the expression and find the period of the subsequent motion.
(10 marks)
(c) How does the motion change if the oscillating system is immersed in water?
(2 marks)
5.
(95-IIB-1b)
(b) A rubber ball is dropped freely from a certain height onto a horizontal floor. It rebounds to
AL-Essay-SHM / p.3
the same height after each bounce. In terms of the force acting, state two ways in which
the motion, although periodic, differs from simple harmonic motion.
(4 marks)
6.
(98-IIB-1)
(a) Explain what a simple harmonic motion (s.h.m.) is, and why it is called an isochronous
oscillation.
(2 marks)
(b) A light spring of force constant k is connected to a block of mass m on a frictionless
surface inclined at an angle  to the horizontal. The block is displaced from its
equilibrium position O and then released. Suppose at a certain instant the displacement of
the block from the equilibrium position is  as shown.
(i)
In terms of the symbols given, express the initial extension eo of the spring when the
block is at its equilibrium position. Show that the period of oscillation of the
subsequent motion is independent of the angle of inclination  .
(ii) The motion of the block can be described by the equation x  A cos 2f t where t
is the time. State the physical meaning of the quantities A and f . On what factor(s)
does each of these quantities depend?
(iii) Sketch, for one cycle, three separate graphs to show how the displacement, velocity
and acceleration of the block vary with time. Comment on their phase relationship.
( No mathematical derivation is required.)
(8 marks)
(c)
The above figure shows a simple pendulum which consists of bob suspended by a light,
inextensible string of length L from a fixed point. If the bob is slightly displaced to one
side and then released, it will perform s.h.m. The set-up can be used to measure the
acceleration due to gravity g.
(i) Which force provides the restoring force for the bob to perform s.h.m. ?
(ii) Give Two reasons why a small spherical heavy bob is usually used in the
experiment.
AL-Essay-SHM / p.4
L
. Describe briefly
g
how the gravitational acceleration g can be determined from this experiment using a
graphical method. State TWO precautions that should be taken in this experiment.
(6 marks)
(iii) The period of oscillation of the pendulum is given by T  2
7.
(02-IIB-3a,b)
(a) (i) Explain the meaning of forced oscillation and resonance. State the phase
relationship, in the case resonance, between the driving force and the
displacement of the system that is being driven.
(ii) Sketch graphs to show how the amplitude of forced oscillation depends on the
driving frequency when the damping is (I) light and (II) heavy. State and explain
TWP considerations in the design of the suspension of a car by referring to the
graphs sketched.
(b) Describe a practical example of resonance which is (i) acoustic and (ii) electrical. For
each example, discuss the factor(s) affecting the resonant frequency.