A torsion spring inside a mechanical watch applies a torque on the balance wheel that…

PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion

Problem

A torsion spring inside a mechanical watch applies a torque on the balance wheel that varies with angular position according to

\tau(\theta) = \tau_{\max}\sin(2\theta)

, where

\tau_{\max} = 1.2\times10^{-3}\,\text{N}\cdot\text{m}

. The balance wheel is displaced from its equilibrium position (

\theta = 0

) to

\theta = \pi/4\,\text{rad}

and then released. (a) Derive an expression for the work done on the balance wheel by the torsion spring as the wheel is displaced from

\theta = 0

to an arbitrary angle

\theta

. (b) Determine the numerical value of the work stored in the spring when the wheel reaches

\theta = \pi/4\,\text{rad}

. (c) The average torque during the displacement can be defined as

\tau_{\text{avg}} = \frac{\text{work}}{\text{angular displacement}}

. Calculate

\tau_{\text{avg}}

for the motion described and compare it to

\tau_{\max}

. Explain why

\tau_{\text{avg}}

is smaller than

\tau_{\max}

. (d) If friction inside the watch removes

3.0\times10^{-4}\,\text{J}

of energy each oscillation, determine whether the energy stored at

\theta = \pi/4\,\text{rad}

is sufficient for the balance wheel to return to its equilibrium position on the first swing. Justify your answer quantitatively.