## Thursday, December 4, 2014

### Problems on Simple Pendulum with Solutions

A simple pendulum is a device which execute simple harmonic motion and whose time period depends on the acceleration due to gravity at a given place.

While we are solving the problems basing on the simple pendulum we shall understand that the time period of a simple pendulum depends on the length of the pendulum as well as the acceleration due to gravity.

The distance between the point of suspension and the Centre of mass of the Bob is called as the length of the pendulum.

The acceleration due to gravity at a place depends on the location.
If any of the external forces are acting on the Bob other than acceleration due to gravity,then we have to take them also into consideration while we are deriving any equation for the time period of a simple pendulum.Taking all these things into consideration some problems are solved solutions are also given below.

Problem and solution

If the bob of the simple pendulum who is made up of iron is replaced a wooden one how it’s time period is going to be affected?

Time period a simple pendulum depends on the length of the bob and is independent of the mass of the bob. Even if you make the bob empty, the time period is  going to remain same.

Problem and solution

A simple pendulum is having a Hallow bob with a certain time period.

1.If the Hallow bob is filled with water.How it’s time period will be affected?

2.If a small puncture is made at the bottom of the bob and water is leaking out drop by drop how it’s time period is affected?

In solving the problem, we have to analyse one basic point that though the time period is not going to depend on the mass of the bob it is going to depend on the length of the pendulum.

When it is filled with water its time period is not going to be affected in any manner. But when a small hole is made at the bottom,water starts leaking out drop by drop and hence the lower part of the bob becomes heavier and hence centre of mass shifts to the lower  part.

Thus the length of the pendulum as well as the time period of the pendulum increases. Anyway this is not going to happen forever. Until the  last a water drop this can happen but once if all water is emptied, the centre of mass comes back to its usual value and hence the time period also will come back to its personal value.

Graphs of the simple pendulum

We can draw simple pendulum graphs taking length on the x-axis and the time on the y-axis as well as length on the x-axis and the square of the time on the y-axis as shown below.

We can also draw both the graphs together in a single graph basing on which we can calculate that at a particular point of length of the pendulum where two graphs are going to intersect with each other.

Basing on the graph we can calculate that this happens at a particular time of one second and hence the corresponding length of the pendulum is 1/4 th of the meter .

A pendulum whose time period is two seconds is called a seconds pendulum. On the earth it can be calculated as approximately equal to 1 m. This is generally taken as a reference pendulum in many of the cases.

Problem and solution

A simple pendulum is having a bob made up of a material with specific density. If this bob is completely immersed in a fluid with specific density, what will be the impact on  the time period?

While solving this problem we shall first of all ignore the viscous forces that are acting. In general we are used to ignore even the buoyant force acting on the bodies assuming that the pendulum is in vacuum.

But it is given in the problem that bob is immersed in a medium,so we shall consider the up thrust acting on the body.Therefore the resultant effect due to  due to gravity and upthrust has to be taken into consideration as shown.

Time period of a simple pendulum in a lift

When the lift is moving in the upward direction the effective acceleration on the body increases and hence the time period decreases.

When the lift is moving in the downward direction the effective acceleration decreases and hence the time period increases.

When the lift  with simple pendulum is moving horizontally with a acceleration,the lift becomes non inertial frame of the reference where we cannot apply the Newton laws of motion directly. To apply the Newton laws of the motion on the pendulum we shall imagine a pseudo-force on the pendulum as shown below.

Problem and solution

If a freely falling body takes a time of five seconds to fall a distance of 150 m at a place. what shall be length of the pendulum so that it can be the seconds pendulum even in that place?

While solving this problem first of all we have to find out the acceleration due to gravity at a place using the concept of the freely falling body. We can further apply the value of the acceleration that we have got in the equation of the time period as shown.

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