1. The document discusses simple harmonic motion (SHM), which is a special type of vibratory motion where the acceleration is proportional to and directed towards the displacement from the equilibrium position. 2. Examples of systems that exhibit SHM include a mass attached to a spring, a simple pendulum, and the motion of a swing. 3. Key characteristics of SHM are that the body oscillates about the mean position, its acceleration is directed towards the mean position and is proportional to the displacement, and its velocity is maximum at the mean position and decreases to zero at the extremes.

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Module # 16
Simple Harmonic Motion
There is a special type of vibratory motion known as simple
harmonic motion and it plays an important role in wave motion.
The type of vibratory motion in which the acceleration is
proportional to the displacement and always directed towards the
equilibrium position (mean position) is called simple harmonic
motion (S.H.M). Note again that equilibrium position and mean
position are the same thing. Hence, the vibrating motion of a
mass attached to a spring is known as simple harmonic motion.
Alternatively, in executing a simple harmonic motion, the
magnitude of the acceleration of a vibrating body is always
directly proportional to the displacement from the mean position
and direction of acceleration is always directed towards the mean
position.
Examples of S.H.M
(a) Motion of simple pendulum.
(b) Motion of a swing.

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(c) Motion of a body hanging from a spring.
(d) Motion of a body attached to one end of a spring whose
other end is attached to a fixed support and is placed horizontally
on a frictionless smooth surface.
(e) The motion of the projection of a particle moving round a
circle with uniform speed.
(f) The vibratory motion of a stretched string of musical
instrument like violin, sitar and sarangi.
(g) The motion of an elastic metallic strip, held vertically in a
rigid support with a heavy mass attached to its free end.
(h) Motion of pendulum of clock.
Characteristics of S.H.M
The body executing simple harmonic motion possesses following
characteristics: -
(1) In SHM, the body executes to and fro motion about its mean
position.
(2) Its acceleration is always directed towards the mean
position.
(3) The magnitude of acceleration is always directly proportional
to the distance of the body from the mean position, i.e., its

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acceleration is zero at mean position and maximum at the
extreme position.
(4) The velocity of body is maximum at the mean position which
decreases to zero as the body reaches the extreme position.
Explanation
Consider a block of mass m attached to one end of a spring and
placed at a point "o" called equilibrium point, on a frictionless
horizontal surface. If we apply an external force Fext to displace
the block to the right through the distance x, the spring will apply a
restoring force F on the block towards left. This restoring force F
at any instant is equal and opposite to the applied external force
Fext.
By Hooke's law, the applied external force Fext is given by
Fext  x
OR
Fext = k x _____ [1]
where K is a constant of proportionality called spring constant. Its
unit is N/m. Since the restoring force F is equal and opposite to
the applied external force Fext, therefore, we can write
F = -k x _______________ [2]

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If xo is the maximum displacement of the block at a point A, when
Hooke's law is still valid, then, we can write the restoring force as
F = -k xo ______________ [3]
We know from Newton's second law of motion that the net force
on an object is equal to the product of its mass m and
acceleration a.
Therefore, for displacement x
F = ma = - K x
OR
a = -k/m x ______ [4]
This is the basic equation of motion for an object undergoing
simple harmonic motion.
Since k and m are constants, therefore, we can write Eq. [4] as
a = - (constant) x
OR

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a  - x
OR
acceleration  (-) displacement
This type of motion of an object in which acceleration is
proportional to its displacement and the acceleration is always
directed towards the equilibrium position is called simple
harmonic motion. The oscillating object is called a simple
harmonic oscillator.
Vibration
One complete round trip of a body about its equilibrium position,
that is, from A to A and back to A is called one vibration.
Time Period (T)
The time taken to complete one vibration is called time period. In
this case,
____
T = 2πm/k
Frequency (f)

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The number of vibrations completed by a vibrating body in one
second is called frequency.
OR
Frequency is the reciprocal of time period, f = 1/T
For mass attached to a spring,
___
f = 1/T = 1/2πk/m
Displacement
Displacement of a vibrating body at any instant is its distance
from the equilibrium position at that instant. It is denoted by x.
Amplitude
The maximum displacement of the body on either side of its mean
position is called amplitude x0.
Basic Conditions for a Frictionless
System to Execute S. H. M
The basic conditions for a system to execute S.H.M. are:
(1) The system must have inertia.
(2) There must be an elastic restoring force acting on the
system.

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(3) The system must obey Hook's law.
(4) The acceleration of the system should be proportional to the
displacement (from the mean position) and must always be
directed towards the mean position.
Simple Pendulum
A simple pendulum is defined as a small heavy body suspended
by a light inextensible string.
Fig: Simple Pendulum
It consists of a small ball (called bob) suspended by a weightless
and inextensible string from a fixed frictionless support. If we
displace the bob from its mean position O to a new position A and
release it, it will move towards O under the action of gravity. It will
not come to rest at O but due to inertia will continue to move
towards a point B. As the bob moves from O to B against the
force of gravity, therefore, its velocity decreases and becomes
zero at the point B. The bob will then move back from extreme
position B to mean position O under the action of gravity. Its

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velocity continues to increase and becomes maximum at mean
position O. It will not stop but will continue to move towards
extreme position A due to inertia. This process is repeated and
the bob continues to vibrate between two extreme positions A and
B. The motion of bob from A to B and back to A is called a
vibration. The time taken to complete one vibration is called time
period and the displacement between O and A or O and B is
called its amplitude. At the lowest position O, the P.E. of the bob
is minimum while its K.E. is maximum. It means that the velocity
of the body is maximum at the mean position O and minimum
at an extreme position A or B.
From this discussion, we have observed that the acceleration of a
simple pendulum is directly proportional to the displacement from
the mean position and is directed towards it. Therefore, the
motion of pendulum is simple harmonic motion.
Time Period
The time taken to complete one vibration is called the time period.
If the amplitude is small, then the time period is given by:
___
T = 2π  ℓ/g
Where ℓ = length of the pendulum, g = Acceleration due to Gravity

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Natural Frequency & Natural Time
Period of a Simple Pendulum
When a simple pendulum is displaced from its mean
position, then, it oscillates with a certain time period T and it will
have a frequency f = 1/T. Whenever, this pendulum is disturbed,
then, it will always vibrate with the same frequency and not with
any other frequency. Therefore, this frequency f is known as the
natural frequency of that pendulum and the time period T as its
natural time period.
Phenomenon of resonance is very often employed to
determine the natural frequencies of the various bodies. All that is
to be done is to apply a small periodic force of known frequency
to the body. If the body begins to vibrate, then, the natural
frequency of the body is equal to the known frequency of the
applied force.
Factors upon which Time Period depends
We know that time period of a simple pendulum
___
T = 2π  ℓ/g

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is quite independent of the mass of the body. Therefore,
pendulums made from bobs of different masses will have the
same time period.
__
As T   ℓ
It means that time period increases with increase in length and
vice versa.
___
Also, T   1/g
It means that as g increases, T decreases and vice versa.
Therefore, the time period of simple pendulum will be smaller at
sea level than that on the mountains. Hence, a pendulum giving
correct time at Karachi will not remain accurate at Murree
because the value of g is different for these places.
Applications of Simple Pendulum
1 To determine Length of a Wire hanging from a Tower
We know that the time period of a simple pendulum is given by
___
T = 2π  ℓ/g

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If we attach a small bob to the lower end of the wire, then, it will
become a simple pendulum and we can determine the time period
of this pendulum by setting it into SHM. Now, by applying the
above formula, we can calculate the length of the wire.
2 To determine the Value of 'g'
With the help of a simple pendulum, we can determine 'g' at a
place, since, the values of ℓ and T can be measured directly. If
proper precautions are taken, the value of g can be determined
very accurately.