Wednesday, December 3, 2014

Time Period of Simple pendulum

Simple pendulum  is a device who can execute simple harmonic motion. It consists of a metal bob of heavy in mass but point in size. Obviously it shall have a higher density. We can define simple pendulum as a heavy and point sized  mass suspended from a rigid support with the help of a in extensible string.

We can show that simple pendulum is in simple harmonic motion and we can also derive a equation for the time period of it as shown below. Consider that we have applied a small force therefore the bob is displaced by a small angle  with the vertical. At that given instant, in the string of length l, there is a force acting towards the point of suspension and that is called as tension. Being the mass is having a gravitational influence on it its weight is always acting in the downward direction. We can resolve the weight into components as shown just because it is a  vector .

It can be notice that this is small vibratory motion is a part of bigger rotatory motion and hence there must be some torque acting on it. We can calculate this turning effect of force using by both the components as shown.

Here we are getting a term called moment of inertia which is similar to the mass of  a body who is in translatory motion. As mass is the deciding factor to identify that how difficult it is to put a body translatory motion, in rotatory motion we are having a similar physical quantity called a moment of inertia. 

If moment of inertia is more it will be difficult to put the body in rotatory motion and vice versa.

The further simplification of the equation is as shown below.

We can identify that the time period of a simple pendulum is independent of the mass of the bob. The length of the pendulum is the distance between the point of suspension and  the centre of mass of the bob.

Therefore though the time period is independent of the mass it depends on the length of the pendulum.

Problem and solution

If the length of a simple pendulum is increased by 69% of its time period is affected?

While solving this problem we can assume that the experiment is done at the same place and hence acceleration due to gravity is constant. Therefore the length of the pendulum under the Squire route is directly proportional to the time period. The problem can be solved as shown below.

Problem and solution

If the length of the pendulum is changed by 5% calculate how it’s time period is affected?

Solving of this problem shall be done differently  from the previous problem. It is simply because he is a variation in the length is a small percentage and hence we can apply approximations concept rather than calculating in lengthy  way of a previous problem.

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