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.
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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
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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:
Weight (mg) of the bob.
Tension T in the string along with OP.
Now, resolving mg into two rectangular components, we get:
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mg CosӨ acts along with PA, opposite to T.
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
Weights
Stopwatch/Timer
Tape
Scale
Straw
Table
Procedure
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.
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.
Let the pendulum settle.
Now, pull the bob about one inch and leave it gently. Make sure to make the pendulum swing in a fixed arc.
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.
Increase the length of the pendulum to 7 inches and the weight of the bobs, and
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.
FAQs on Changing the Period of a Pendulum
1. What is the formula used to calculate the time period of a simple pendulum?
The formula for the time period (T) of a simple pendulum is T = 2π√(L/g). In this formula, T represents the time period in seconds, L is the effective length of the pendulum in meters, and g is the acceleration due to gravity, which is approximately 9.8 m/s² on Earth.
2. What are the two primary factors that can change the period of a simple pendulum?
Based on its formula, the time period of a simple pendulum is affected by only two factors:
- The effective length (L) of the pendulum.
- The acceleration due to gravity (g) at the pendulum's location.
For small oscillations, the period is considered independent of the mass of the bob and the amplitude of the swing.
3. How can you experimentally change the period of a pendulum?
To change a pendulum's period, you must alter its length. To make the pendulum swing slower and increase its time period, you need to increase its length. Conversely, to make it swing faster and decrease its time period, you must shorten its length. This is the fundamental principle used to regulate pendulum clocks.
4. Why does increasing a pendulum's length cause its time period to increase?
Increasing a pendulum's length increases its time period because it affects both the path and the acceleration. A longer pendulum has a greater distance to travel for each swing. More importantly, the restoring force that pulls the bob back to its central position is a component of gravity. For a longer pendulum, this force produces a smaller tangential acceleration, causing the bob to move more slowly. This slower movement over a longer path results in a longer time to complete one full oscillation.
5. Does the mass of the pendulum bob affect its time period? Explain why.
No, the mass of the pendulum bob does not affect its time period. This is a common misconception. While a heavier bob experiences a greater gravitational force, it also has a proportionally greater inertia (resistance to a change in motion). These two effects precisely cancel each other out in the equation of motion, making the period independent of mass.
6. If a pendulum clock is running slow, how would you adjust it to keep the correct time?
If a pendulum clock is running slow, it means its time period is too long. To correct this, you need to make the pendulum swing faster, which means you must decrease its time period. Since the period is directly proportional to the square root of the length (T ∝ √L), you must shorten the length of the pendulum rod to make it keep the correct time.
7. What is the difference between an ideal simple pendulum and a real pendulum we use in experiments?
An ideal simple pendulum is a theoretical concept with specific assumptions that differ from a real pendulum:
- Mass: An ideal pendulum has a point mass (the bob), while the string is massless. A real pendulum's bob has volume, and the string has some mass.
- String: The string of an ideal pendulum is considered perfectly inextensible and flexible. Real strings can stretch slightly.
- Friction: An ideal pendulum assumes no air resistance or friction at the pivot point. A real pendulum is always affected by these dissipative forces, which causes its oscillations to eventually die down.
8. What would happen to the time period of a pendulum if you took it from Earth to the Moon?
The time period of the pendulum would significantly increase. The acceleration due to gravity on the Moon is about 1/6th of that on Earth. Since the time period (T) is inversely proportional to the square root of gravity (g), a much smaller value of 'g' on the Moon would result in a much longer time period. The pendulum would swing back and forth very slowly.
