Simple Harmonic Motion
Elasticity
• Many objects change shape when a force is applied
to them, e.g. elastic band. When the force is
removed the object may return to its original shape,
i.e. object is said to be elastic.
• If the force applied is too great and object remains
permanently strained it has exceeded its elastic
limit.
• The force trying to pull the object back into its
original position is the restoring force. This force
is directly proportional to the displacement.
Definition
• Hooke’s law states that when an
object is bent, stretched or
compressed by a displacement ‘s’,
the restoring force ‘F’ is directly
proportional to the displacementprovided the elastic limit is not
exceeded.
F=–ks
where k = elastic constant
• The above equation is
known as Hooke’s Law
(after Robert Hooke
(1635-1703), an
inventor, philosopher,
architect, ...)
Simple Harmonic Motion (S.H.M.)
• Position O: at rest; so no
force
• Force opposing the
displacement
•
•
Position O is called the equilibrium position. If
pulled beyond O it vibrates up and down. When
doing this particle can be said to be moving in
simple harmonic motion.
Definition: A body is said to be moving with
simple harmonic motion if:
1. Its acceleration is directly proportional to its
distance from a fixed point on its path.
2. Its acceleration is always directed towards that
point.
• Let ‘a’ be the acceleration and let O be the fixed
point on its path. Let ‘s’ be displacement of particle
from O. since ‘a’ is always opposite in direction to
2
‘s’, the equation defining S.H.M. is: a   s
• The negative sign shows ‘a’ and ‘s’ are always in
opposite directions.
• Any system obeying Hooke’s law will execute
S.H.M.
• If system obeys Hooke’s law then:
F=–ks
 ma=–ks
(since F = m a)
 ks  k
k
2
2
a

s, where  
as a   s
m
m
m
• system moves in S.H.M.
Examples
• Tides moving in and out approximately every 6
hours.
• For a small angle of swing, a pendulum moves with
SHM.
• Each prong on a tuning fork vibrates with SHM.
• Mass on a vibrating spring is SHM.
Uses
• Car shock absorbers. The work due to dampening
the simple harmonic motion.
Terms used to describe S.H.M.
If a body is moving in S.H.M. then:
• A cycle or an oscillation is the movement from A to
B and back again.
• The periodic time or period ‘T’ of a particle
executing S.H.M. is the time of one complete
oscillation. ‘T’ is measured in seconds.
• The frequency ‘f’ is the number of cycles occurring
per second. Frequency is measured in cycles per
second. One cycle per second is called one hertz
(Hz).
• Since ‘T’ is time for one oscillation and ‘f’ is the
number of oscillations occurring per second we see
that:
1
1
T
f
and
f 
T
• The amplitude is the greatest displacement that the
particle has from equilibrium position.
• As the body moves with S.H.M. its energy changes
continually from potential to kinetic energy and
back again. At A and B, potential is maximum and
kinetic is zero. At O, potential is zero and kinetic is
maximum.
Formula for Period of S.H.M.
• If particle moves with S.H.M. whose eqn. is:
a   s
2
then the period ‘T’ of the motion is given by:
T
2

Simple Pendulum
• This consists of a small mass called
a bob attached to a fixed point by a
light inelastic string. The length of
the pendulum ‘l’ is the distance
from the fixed point of suspension
to the centre of the bob. When bob
is set swinging the time taken to
swing from A to B and back again
is the period ‘T’ of the pendulum.
Due to air friction, the amplitude of
the oscillations decrease as time
goes on. However as amplitude
decreases, the period remains the
same. Galileo discovered this fact.
• For a small angle of swing (i.e. not more than 5o
from the vertical at either side) a simple pendulum
moves with S.H.M.
• The period ‘T’ of a pendulum of length ‘l’ is given
by:
l
T  2
g where g = acceleration due to gravity
Measuring acceleration due to gravity
by a pendulum
• Squaring both sides of
l
T  2
g
l
 T  4
g
2
2
l
 g  4 2
T
2