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Coupled Pendulum Viva Questions

Before beginning, I recommend that you understand Pendulum by clicking here, since questions are asked from it.

What is a coupled pendulum?

A coupled pendulum is a system of two or more pendulums connected by a spring or other coupling mechanism. This connection allows for energy transfer between the pendulums, leading to interesting dynamic behaviours like beat phenomenon.

What is a beat?

Beat is produced when two waves of nearby frequencies superimpose when they travel in the same path. It is the number of beats produced in one second.

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What is an angular frequency?

Angular frequency measures angular displacement per unit time. Also known as radial or circular frequency.

What is angular momentum?

Angular momentum represents the product of a body’s rotational inertia and rotational velocity about a specific axis.

Rarely asked Questions
What is damping factor?

In a coupled pendulum experiment, the damping factor measures how quickly oscillations decrease due to energy loss from friction or air resistance.

The damping factor is often represented by the Greek letter ζ(zeta) and is a dimensionless quantity. It can be classified into three categories:

  • Underdamping (ζ<1): The system oscillates with gradually decreasing amplitude.
  • Critical Damping (ζ=1): The system returns to equilibrium as quickly as possible without oscillating.
  • Overdamping (ζ>1): The system returns to equilibrium without oscillating, but more slowly than in critical damping.(More frequently asked)
What is degree of freedom?

The minimum number of coordinates needed to completely describe the motion of a pendulum.

For example- A simple pendulum that is constrained to pivot in one place can have its position specified by a single coordinate (usually angular displacement) and has only one natural frequency of vibration.

What are dynamic oscillations?

Dynamic oscillations in a coupled pendulum experiment refer to the periodic motion caused by the interaction and energy exchange between the pendulums. These oscillations include characteristics like amplitude, frequency, phase, and damping.

Derivation of natural frequencies in the experiment:
How to get

    \[ \omega_1 = \sqrt{\frac{g}{L}} \]

We know, \theta_1 = A_1 \cos \omega t … eqn(1)

We differentiate eqn(1) to get \frac{d\theta_1}{dt} and differentiate eqn(1) twice we get \frac{d^2 \theta_1}{dt^2}.

After putting these values in the equation:

    \[ ML^2 \frac{d^2 \theta_1}{dt^2} = MgL\theta_1 - Ka^2 (\theta_1 - \theta_2) \]

And putting \theta_1 = \theta_2 for the same phase, we get:

    \[ \omega_1 = \sqrt{\frac{g}{L}} \]

How to get

    \[ \omega_2 = \sqrt{\frac{g}{L} + \frac{2ka^2}{ML^2}} \]

Similarly for \theta_2 = A_2 \cos \omega t … eqn(2)

We differentiate eqn(2) to get \frac{d\theta_2}{dt} and differentiate eqn(2) twice we get \frac{d^2 \theta_2}{dt^2}.

After putting these values in the equation:

    \[ ML^2 \frac{d^2 \theta_2}{dt^2} = MgL\theta_2 + Ka^2 (\theta_1 - \theta_2) \]

And putting \theta_1 = -\theta_2 for opposite phase, we get:

    \[ \omega_2 = \sqrt{\frac{g}{L} + \frac{2ka^2}{ML^2}} \]

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