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.
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.
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.
Aim: To change the period of a pendulum
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.
The number of swings divided by the time taken gives you the period of a pendulum.
Repeat the trials and record the results.
Q1: Why is the Simple Pendulum called simple?
Ans: A simple pendulum is also called the mathematical pendulum or the simple gravity pendulum. It is assumed that all the mass is the bob; the rest of the pendulum is massless. This gives the easiest way to look at its harmonic motion. That’s why it is called the simple pendulum.
Q2: Could we ever construct a True Simple Pendulum?
Ans: Practically, it is impossible to create an ideal simple pendulum because we can’t have a point mass and heavy solid metallic sphere (bob) tied to one end of a long unspun cotton thread and suspended from rigid support to serve the purpose of a simple pendulum.
Q3: How does a Pendulum Swing?
Ans: A pendulum makes an oscillatory motion under the influence of gravity.
As we swing it to one end from its mean position, it starts oscillating because of the force of gravity.
Q4: What distinguishes a Simple Ideal Pendulum from a Real Pendulum?
Ans: An ideal pendulum comes with a concept that a point mass is to be suspended by a weightless, flexible string while a real pendulum is a compound pendulum in which a body of any shape is suspended by a string to oscillate.