Question

Define Time period of a pendulum.

Answer 1

Hint: Time period of anything is the amount of time required for it to complete one oscillation. Usually when we have angular velocity we can get frequency from it and when we get frequency the inverse of that frequency gives us the time period. If an object is executing simple harmonic motion then we find the restoring force first and then time period from that.
Formula used:
$T = 2\pi \sqrt {\dfrac{L}{g}} $

Complete answer:
Frequency is the number of waves passing through a point within one second. If we take inverse of that frequency, we will get the time period. In case of satellites revolving around any planet we get angular velocity of rotation of that satellite by equating the centripetal force of that satellite to the gravitational force of attraction between satellite and that planet and then we will find out the frequency and we will take the inverse of frequency to get the time period.

In case of simple pendulum time period is the time required for the simple pendulum to do one complete oscillation i.e if a pendulum starts from the mean position it will travel all the way back to its mean position after completing one total oscillation. The amount of time required for this is called the time period of simple pendulum.

We will derive it by giving small displacement to the simple pendulum from the mean position then component of gravity tries to bring it back to the mean position and we equate that force to the restoring force and from that we will get the angular velocity and finally we get the time period from that.

We get that time period as $T = 2\pi \sqrt {\dfrac{L}{g}} $ where L is the length of the string and g is the acceleration due to gravity which is constant in this case.

Note:
The formula which we had derived is valid only for small displacements of the pendulum from its mean position and when acceleration due to gravity is constant and when length of the string is very negligible compared to the radius of the earth. If the length of the string is comparable to the radius of earth then we get a different formula.