Simple pendulum consists of a point mass suspended by inextensible weightless string in a uniform gravitational field.
Simple pendulum can be set into oscillatory motion by pulling it to one side of equilibrium position and then releasing it.
In case of simple pendulum path ot the bob is an arc of a circle of radius l, where l is the length of the string.
We know that for SHM F=-kx and here x is the distance measured along the arc as shown in the figure below.
When bob of the simple pendulum is displaced from its equilibrium position O and is then released it begins to oscillate.
Suppose it is at P at any instant of time during oscillations and θ be the angle subtended by the string with the vertical.
mg is the force acting on the bob at point P in vertically downward direction.
Its component mgcosθ is balanced by the tension in the string and its tangential component mgsinθ directs in the direction opposite to increasing θ .
Thus restoring force is given by
F=-mgsinθ (19)
The restoring force is proportional to sinθ not to The restoring force is proportional to sinθ, so equation 19 does not represent SHM.
If the angle θ is small such that sinθ very narly equals θ then above equation 19 becomes
F=-mgθ
since x=lθ then,
F=-(mgx)/l
where x is the displacement OP along the arc. Thus,
F=-(mg/l)x (20)
From above equation 20 we see that restoring forcr is proportional to coordinate for small displacement x , and the constant (mg/l) is the force constant k.
Time period of a simple pendulum for small amplitudes is
Corresponding frequency relations are
and angular frequency
ω=√(g/l) (23)
Notice that the period of oscillations is independent of the mass m of the pendulumand for small oscillations pperiod of pendulum for given value of g is entirely determined by its length.
(c) The compound pendulum
Compound pendulum is a rigid body of any shape, capable of oscillating about a horizontal axis passing through it.
Figure below shows vertical section of rigid body capable of oscillating about the point A.
Distance l between point A and the centre of gravity G is called length of the pendulum.
When this compound pendulum is given a small angulr displacement θ and is then released it begins to oscillate about point A.
At angular displacement θ its center of gravity now takes new position at G'.
Weight of the body and its reaction at the support constitute a reactive couple or torque given by
τ=-mg G'B
=-mglsinθ (24)
Equation 24 gives restoring couple which tends to bring displaced body to its original position.
If α is the angular acceleration produced in this body by the couple and I is the moment of inertia of body about horizontal
axis through A then the couple is
Iα=-mglsinθ
if θ is very small then we can replace sinθ≅θ, so that
α=-(mgl/I)θ (25)
From above equation (25) we se that pendulum is executing Simple Harmonic Motion with time period