Measurement of Time by a Simple Pendulum

example

Find frequency and time period of a simple pendulum

Example: A simple pendulum hangs from the ceiling of a car. Time period of pendulum when the car is at rest was found . Now if the car moves uniformly in a large circle, pendulum stays in equilibrium with respect to car at an angle of with the vertical. Find the new time period of the pendulum for small oscillations ? (use )

Solution:
We know that time period of simple pendulum is,

Now it makes angle  when car is moving in a circle. Therefore 
component of acceleration due to net force acting on the string will be

New Time period, , will be


Also,
--------  
Solving and , we get




formula

Time period for a physical pendulum

Time period of Physical pendulum:
where is length of pendulum in cm.

example

Period of a pendululm of infinite length

Time period of a pendulum of infinite length =
Where tends to infinity.

definition

Define Time period

The time taken by the wave for one complete oscillation for the density or pressure of the medium is called the time period. It is measured in seconds.

e.g. The time period of the minute hand is 60 minutes.

example

Calculate time period

Problem:
A common hydrometer has and specific gravity marks cm apart. Calculate the time period of vertical oscillations when it floats in water. Neglect resistance of water.
Solution:
lets say is the length of the hydrometer when it is dipped in water and is the cross section area of the hydrometer, then cm length of the hydrometer will be dipped when it is placed in liquid of specific gravity 0.8. So we have 
weight of hydrometer=water displaced = liquid (0.8 s.g.) displaced

And time period of a floating cylinder (hydrometer )is given by where is the length of the cylinder (hydrometer) dipped in water. 


diagram

Interpret length vs time period graphs of a simple pendulum

The following graph shows variation of time period with the length of pendulum. 

definition

Factors affecting the time period of a simple pendulum


where
Time period, length, acceleration due to gravity
1. The time period of oscillation is directly proportional to to the square root of its effective length.
2. The time period of oscillation is inversely proportional to the square root of acceleration due to gravity.
3. The time period of oscillation does not depend on the mass or material of the body suspended.
4. The time period of oscillation does not depend on the extent of swing on either side.

diagram

Interpret length vs time period graphs of a simple pendulum

Time period of Pendulum

hence, 

The graph between time period and length becomes parabola as iterpreted in figure.