Question

Derive an expression for the period of oscillation of a torsion pendulum ?

Answer 1

Answer:

The period of a torsional pendulum T=2π√Iκ T = 2 π I κ can be found if the moment of inertia and torsion constant are known.

Explanation:

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Answer 2

Answer:

The time period of the torsional pendulum $2\pi\sqrt{\frac{I}{k}}$.

Explanation:

• A torsional pendulum is a stiff body suspended by a wire with one end fastened rigidly to the ceiling and around which the body can execute horizontal angular oscillations.
• When the pendulum is at rest, there is no torque acting. The pendulum begins to execute angular SHM when it is given a tiny "twist" and then let free. So,                                        

                                          $\tau \propto (-\theta)$

                                          $\tau =-k\theta$   (1)

Where,

τ=torque acting on the pendulum

θ=angle through which the pendulum is rotated

k=torsional constant

                                     $\alpha=\frac{\tau}{I}$     (2)

α=angular frequency

I=moment of inertia

using equation (1) in equation (2);

                                  $\alpha=\frac{-k\theta}{I}$      (3)

We know that angular frequency is also given as,

                               $\alpha=-\omega^2\theta$     (4)

Equating equations (3) and (4) we get;

                            $-\omega^2\theta=\frac{-k\theta}{I}$

                                $\omega=\sqrt{\frac{k}{I}}$     (5)

And the time period(T) is given as,

                                $T=\frac{2\pi}{\omega}$     (6)

Using equation (5) in equation (6) we get;

                              $T=2\pi\sqrt{\frac{I}{k}}$  

This is the time period of the torsional pendulum.

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