A torsional pendulum consists of a rigid circular disc of moment of inertia suspended…

PHYSICS · Subjective · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

A torsional pendulum consists of a rigid circular disc of moment of inertia

I

suspended by a thin wire that provides a restoring torque

C\theta

, where

C

is the torsional constant and

\theta

is the angular displacement. A damping torque proportional to the angular velocity acts with coefficient

b_t

. (i) Show that the natural angular frequency of small torsional oscillations is

\omega_0 = \sqrt{\dfrac{C}{I}}

. (ii) Derive an expression for the quality factor

Q

of the torsional pendulum in terms of

I

C

and

b_t

. (iii) For

I = 1.5\,\text{kg\,m}^2

C = 0.60\,\text{N\,m\,rad}^{-1}

and

b_t = 0.03\,\text{kg\,m}^2\text{\,s}^{-1}

, calculate

Q

. (iv) If the damping torque doubles while

I

and

C

remain unchanged, determine the percentage decrease in

Q

and comment on the consequent change in the resonance curve.