Changing the Period of a Pendulum

A simple pendulum is the example of bodies executing simple harmonic motion.

A pendulum consisting of a point mass body (bob) suspended by a weightless, inextensible and flexible string from rigid support about which it is free to move back and forth is called an ideal simple pendulum.

[Image will be uploaded soon]

In an equilibrium position, the center of gravity of the bob lies vertically below the point of suspension.Here, O is called the equilibrium or the mean position or the point of oscillation of the simple pendulum. 

The distance between the point of suspension and the point of oscillation is called the effective length of a pendulum.

The Expression for the Period of Oscillation

[Image will be uploaded soon]

Let the mass of bob = m

L = length of the simple pendulum

O1P = s

When the bob is displaced from its mean position by an angle Ө, the forces acting on it at P are:

1. Weight (mg) of the bob.

2. Tension T in the string along with OP.

Now, resolving mg into two rectangular components, we get:

[Image will be uploaded soon]

1. mg CosӨ acts along with PA, opposite to T.

2. mg SinӨ acts along with PB, tangent to the arc O1P, and directed towards O1.

 If the string neither slackens nor breaks, then:

                                      T = mg CosӨ 

The force mg SinӨ tends to bring the bob to its mean position, so the restoring force will be:

                                       F = - mg SinӨ

Here, a negative sign shows that the pendulum shows the property similar to that of inertia. It tries to come back to its mean position.

Since Ө is very small, so, Sin Ө becomes equal to Ө = Arc (O1P)/L = s/L 

F = - mg Ө = - mgs/L…(1)

From eq(2),  we come to know that Force, F α displacement (s).

This F is directed towards the mean position O.

If the pendulum is left after stretching it, it starts executing an S.H.M.

So, in S.H.M., the restoring force, F = - ks..(2)

Combining (1) and (2), we get:

Spring factor, k = mg/L

Here, inertia factor = mass of bob = m 

Now, time period = 2π √inertia factor/spring factor = 2π√m/mg/L.               

So, we get the formula for a period as;                           

So, T is α √L and √1/g

Now, let’s understand how we can change the period of a pendulum.

Changing the Period of a Pendulum

Equation (3), states that T α √L, which means T increases with an increase in the effective length L.

Case 1: If the effective length L of the pendulum is very large which is comparable to the radius of the earth R, then T can be shown as:

                                    T = 2π√R/(1+R/L)g

Case 2: Now, if this length increases to the length greater than R or reaches to infinity

 (L → ∞), then the period becomes:

                                    T = 2π√R/g 

Putting R = 6.4 𝑥 10⁶m, and g = 9.8 ms^-2,

                                   T =  2π√6.4 𝑥 10⁶/9.8 = 84.6 minutes.

Case 3: If this length = R, then,

                                   T = 2π√R/2g = 2π√6.4 𝑥 10⁶/2 x 9.8

                                      = 60 minutes

This is how we keep on changing the period of a pendulum.

So, how can you increase the period of a pendulum?

Let’s look at these cases:

Case 4: T α √1/g

When the value of g decreases, the value of T increases with the decrease in the value of g by taking the pendulum in hilly areas. 

However, the pendulum clock slows down.

Case 5: With the rise in temperature of the pendulum, the effective length of a pendulum increases, along with the period.

                           T/T’ = √L/L’ = (1 + αӨ/2) 

% increase in time period = 50 αӨ.

Case 6: If the pendulum is made to oscillate in a fluid of density ρ0, where ρ0 < ρ, then, 

                       T/T’ = √g/g’ = √ρ/ρ - ρ0 > 1 (as ρ > ρ0)

So, the period increases with a decrease in g.

This is how you can increase the period of a pendulum.

Pendulum Swing Experiment

Aim: To change the period of a pendulum

Apparatus Required

1. Weights

2. Stopwatch/Timer

3. Tape

4. Scale

5. Straw

6. Table

Procedure

1. Tie a weight (of mg) to a thread and then tie the thread to a straw on a table such that around half of an inch hangs over the edge.

2. Tape the other end of the thread with the table in such a way that the length from the straw’s end to the middle of the weight is 6 inches.

3. Let the pendulum settle.

4. Now, pull the bob about one inch and leave it gently. Make sure to make the pendulum swing in a fixed arc.

5. When you take the bob at one end as soon as you leave, start the stopwatch to count the number of swings made by the pendulum.

6. Increase the length of the pendulum to 7 inches and the weight of the bobs, and

7. Repeat the above procedure.

Result

The number of swings divided by the time taken gives you the period of a pendulum.

Repeat the trials and record the results.