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 2^{nd}
law, F = ma
\ ma = kx =

mg

.x


l

\ a =

g

x = w^{2}x


\

a

= ^{g}/_{l}
(numerically)



l

x

Period T =

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


p^{2}

For g = 9.8 m/s^{2},

l =

98

= 0.9940 m


(3.14)^{2}

Laws of a Simple Pendulum
For
a simple pendulum, T = 2p . Hence,
 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.
 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.



 The period of a simple pendulum is independent of the
mass of the bob.
 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
Ia = kq
\ a =

k

.q


I

a =

d^{2}q

\

d^{2}q

=

k

q




dt^{2}

dt^{2}

I

\

d^{2}q

+

k

q = 0



dt^{2}

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 = = 2p
T =
2p
