MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Simple Harmonic Motion Page 4

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S.H.M. of a Simple Pendulum

Consider a simple pendulum of length l, displaced through a small angle q from its mean position and allowed to oscillate about its mean position.
In the displaced position, the forces acting on the bob are:
(i) Its weight,
mg, acting downwards
(ii) The tension, T, acting along the string

Resolving
mg into components as shown, mg cos q balances T.
mg sin q is unbalanced and acts towards O. It, thus, acts as a restoring force on the bob.

\ Restoring force F = -mg sin q (The negative sign indicates that the direction of force is opposite to the direction of displacement of the bob).

Since
q is very small and is measured in radians,

 Sin q » q = arc = AB = x radius OB l

 \ F = -mg .x i.e. F = -kx where k = mg l l

Thus, the motion of a simple pendulum is simple harmonic in nature.

Expression for Period
By
Newton's 2nd law, F = ma

 \ ma = -kx = -mg .x l

 \ a = -g x = -w2x \ a = g/l (numerically) l x

 Period T = 2p

 \ T = 2p = 2p

\ T = 2p
Seconds Pendulum: A simple pendulum whose period is 2 seconds is called a seconds pendulum.
For a seconds pendulum, T = 2s
\ 2p = 2

 \ l = g p2

 For g = 9.8 m/s2, l = 98 = 0.9940 m (3.14)2

Laws of a Simple Pendulum

For a simple pendulum, T = 2p . Hence,

1. The period of a simple pendulum is directly proportional to the square root of the
length at a given place, i.e., T
µ , if g is constant.
2. The period of a simple pendulum is inversely proportional to the square root of
the acceleration due to gravity for a given length.
 \ T µ 1 , if l is constant.
1. The period of a simple pendulum is independent of the mass of the bob.
2. The period of a simple pendulum is independent of the amplitude of oscillation.

Angular S.H.M.
Definition: The oscillatory motion of a body in which the restoring torque acting on the body acts in a direction opposite to that of the angular displacement of the body from its mean position and whose magnitude is directly proportional to the magnitude of the angular displacement is called angular S.H.M.

Here,
t = -kq
Since
t = I a
I
a = -kq

 \ a = -k .q I

 a = d2q \ d2q = -k q dt2 dt2 I

 \ d2q + k q = 0 dt2 I

This is the differential equation of angular SHM.

Magnet in a Uniform Magnetic Field

Consider a magnetic dipole of moment m suspended in a uniform field of induction B. In the displaced position, restoring torque t acts on the magnet.

t = -mB sin q. (The negative sign indicates that t and q are oppositely directed.)
Now,
t = I a
\ Ia = -mB sin q
a = sin q

Since
q is small and measured in radians,

 Sin q » q. \ a = -mB .q I

 Thus, a µ q since ( -mB ) is a constant. I

\ Bar magnet executes angular SHM.

 Angular acceleration per unit angular displacement = mB I

 Period T = 2p

i.e. T = = 2
p

T = 2p

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