Compound and Torsional pendulum, practice problem, FAQs

You may see an oscillating bob in clocks, which is suspended from a fixed point so that it can swing back and forth under the influence of gravity. This is called a compound pendulum and these clocks are called pendulum clocks.Pendulum has a constant time period so it is used to regulate the movement of the clock. Lets know more about the pendulum.

Table of content

• Compound pendulum
• Time period of compound pendulum
• Variation of time period with length
• Torsional Pendulum
• Practice problem
• FAQs

Compound pendulum

If a rigid body is suspended from a fixed point and it executes oscillations about the axis passing through that point of suspension, then it is known as a compound pendulum or physical pendulum.

At equilibrium, the COM of the rigid body will lie on the vertical line that passes through the point of suspension, so that the torque due to the gravitational force about the point of suspension becomes zero. If we disturb this body and move the CM a little away from the vertical line and release it the body will execute the angular harmonic motion.

Time period of compound pendulum

Suppose that the rigid body is displaced by a small angle θ from the mean position as shown in the figure.Lets the moment of inertia about the suspension O is and distance between suspension and center of mass of body is ,

Therefore, the torque about point O is,

Where minus (-) Sign shows that the torque is restoring

Now, For small ,

As we know for a angular SHM, angular acceleration is given as

On comparing with the above eq

Hence For small angular displacement the rigid body will execute SHM and angular frequency of SHM will be

The time period of SHM is 

Now, using the parallel axis theorem, the moment of inertia of the rigid body about the point of suspension, , can be written as , and if is the radius of gyration about the COM, then . Therefore, the moment of inertia of the rigid body about the point of suspension becomes, .

Therefore, the time period becomes, 

Now, if we compare this expression of time period of a compound pendulum with that of a simple pendulum of length leq of the same time period, then the equivalent length of the simple pendulum will be, 

Therefore, the time period of a physical pendulum is the same as that of a simple pendulum of length .

Variation of time period with length

The radius of gyration, , cannot be changed because the moment of inertia of a rigid body about its COM is invariant, but we can change the length of the pendulum. 

If we differentiate the time period with respect to the length, l, and equate it to zero, we get the following:

Therefore, the time period is minimum at l = k and the minimum value of the time period is, 

. 

The variation of time period is shown in the figure.

It is seen from the figure that except at the minimum time period, , there are always two positions to suspend the physical pendulum for a particular time period.

Torsional Pendulum

Suppose that an extended body is suspended by a light, massless thread.

If the body is rotated about the thread as the axis rotation by a small angle θ, it is observed that the restoring torque generated by the thread of the pendulum is proportional to the angle of rotation (θ), i.e.,

Where is the torsional constant or twisting coefficient

Now, if the angular acceleration be then

There for,

Hence, the extended body executes torsional oscillations with angular frequency 

Therefore, the time period of the torsional oscillation is,

.

Practice problem

Q1. Find the time period of small oscillations for a ring of mass m and radius r suspended through a point on its periphery
Answer: The moment of inertia of the ring about the COM is, . 

Thus, the moment of inertia of the ring about point O is,

, 

The distance of the COM from the point of suspension is,

Therefore, the time period is,

Q2. Find the time period of small oscillations for a uniform square plate of edge suspended through a corner. 

Answer. Since the square plate is suspended at one corner, the distance between the COM and the point of suspension is half of the length of the diagonal of the square. 

Now, the moment of inertia of a uniform square plate about its COM is, 

Therefore, the moment of inertia of the plate about the point of suspension (O) is,

The time period of the oscillation is, 

Q3. Find the time period of small oscillations for a uniform disc of mass and radius suspended through a point away from the center.

Answer.The distance of the COM from the point of suspension is,

The moment of inertia of the disc about the COM is, 

Thus, the moment of inertia of the disc about point O is,

The time period of the oscillation is, 

Q4. A uniform disc having radius is suspended through a small hole in the disc.What is the minimum possible time period of the disc for small oscillations. Also find the distance of the hole from the center so that it has a minimum time period?

Answer. Let the radius of gyration be k.

We know that the moment of inertia of a disc about an axis passing through its COM is,

The minimum possible time period of the disc for small oscillations is,

and the distance of the hole from the center for it to have a minimum time period is,

Ans

FAQs

Question 1. How does the compound pendulum differ from a simple pendulum?
Answer:  The period of a simple pendulum depends only on the length of the pendulum and on the other hand in compound pendulum the object’s mass and how that mass is distributed. 

Question 2. What is the physical pendulum?
Answer: The physical pendulum is any rigid body that is pivoted so that it can oscillate freely. It is the same as a compound pendulum.

Question 3. Which material is used for the clock pendulum?
Answer: Invar is used because of its stability and low thermal expansion.

Question 4. What is the time period of the pendulum clock?
Answer: Time period of the pendulum clock is 2 sec.

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