There are two forces acting on the particle mass hanging from the string (called pendulum bob). One is the gravity (mg), which acts vertically downward. Other is the tension (T) in the string. In equilibrium position, the bob hangs in vertical position with zero resultant force :

Simple pendulum |
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At a displaced position, a net torque about the pivot point “O” acts on the pendulum bob which tends to restore its equilibrium position. In order to calculate net torque, we resolve gravity in two perpendicular components (i) mg cosθ along string and (ii) mg sinθ tangential to the path of motion.

Together with tension and two components of gravity, there are three forces acting on the pendulum bob. The line of action of tension and the component of gravity along string passes through pivot point, “O”. Therefore, torque about pivot point due to these two forces is zero. The torque on the pendulum bob is produced only by the tangential component of gravity. Hence, torque on the bob is :

where "L" is the length of the string. We have introduced negative sign as torque is clockwise against the positive direction of displacement (anticlockwise). We can, now, use the relation “τ =Iα” to obtain the relation for angular acceleration :

Clearly, this equation is not in the form “

Is it possible? Not exactly, but approximately yes - if the angular displacement is a small measure. Let us check out few values using calculator :

```
--------------------------------------------
Degree Radian sine value
--------------------------------------------
0 0.0 0.0
1 0.01746 0.017459
2 0.034921 0.034914
3 0.052381 0.052357
4 0.069841 0.069785
5 0.087302 0.087191
--------------------------------------------
```

For small angle, we can consider "

We have just seen the condition that results from the requirement of SHM. This condition requires that angular amplitude of oscillation should be a small angle.

**Angular frequency**

Comparing the equation obtained for angular acceleration with that of “

There is yet another aspect about moment of inertia that we need to discuss. Note that we have considered that bob is a point mass. In that case,

and

We see that angular frequency is independent of mass. What happens if bob is not a point mass as in the case of real pendulum. In that case, angular frequency and other quantities dependent on angular frequency will be dependent on the MI of the bob – i.e. on shape, size, mass distribution etc.

We should understand that requirement of point mass arises due to the requirement of mass independent frequency of simple pendulum – not due to the requirement of SHM. In the nutshell, we summarize the requirement of simple pendulum that arises either due to the requirement of SHM or due to the requirement of mass independent frequency as :

- The pivot is free of any energy loss due to friction.
- The string is un-strechable and mass-less.
- There is no other force (other than gravity) due to external agency.
- The angular amplitude is small.
- The ratio of length and dimension of bob should be large so that bob is approximated as point.

**Time period and frequency**

Time period of simple pendulum is obtained by applying defining equation as :

Frequency of simple pendulum is obtained by apply defining equation as :