A torsional pendulum consists of a solid disk that can rotate about its central axis…

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

Problem

A torsional pendulum consists of a solid disk that can rotate about its central axis while suspended by a thin wire. The period of small-amplitude oscillations of the system is given by

T = 2\pi \sqrt{\dfrac{I}{\kappa}},

where

I

is the moment of inertia of the disk about the axis and

\kappa

is a constant that characterizes the restoring torque provided by the wire. The disk has mass

0.50\,\text{kg}

and radius

0.10\,\text{m}

. For the purpose of this problem, the disk’s moment of inertia about its central axis is

I = 2.5\times10^{-3}\,\text{kg}\,\text{m}^{2}

. (a) Using dimensional analysis, determine the required SI base units of the constant

\kappa

so that the expression for

T

is dimensionally consistent. Express your answer as

\text{kg}^{a}\,\text{m}^{b}\,\text{s}^{c}

. (b) The measured oscillation period is

1.20\,\text{s}

. Calculate the numerical value of

\kappa

(to two significant figures) that would produce this period for the given disk. Express your answer using the unit structure obtained in part (a). (c) Provide a concise justification that your numerical answer in part (b) indeed yields a time with units of seconds when substituted back into the period equation.

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