Question

A "reference ring" with moment of inertia Ic about the axis shown rests on a solid disk torsion pendulum with a moment of inertia Io. 1. What is the period To for motion with the disk alone? Assume that the torsion constant of the wire is k. 2. What is the period T1 for motion with the combined system (when the reference ring is placed on top of the disk and is rotating with it)? Assume that the torsion constant of the wire is k. 3. Use the expression from 1. and 2. to derive an expression for Io in terms of To,T1, and Ic (you will find it most convenient to solve if you take the ratio of the equation in 1. and 2.)

Answer 1

The period of the disk alone is To = 2π√(Io/k), and the period with the reference ring is T1 = 2π√((Io + Ic)/k). By taking the ratio of T1 to To and rearranging, Io can be expressed as Io = Ic/((T1/To)^2 - 1).

Step-by-step explanation:

When considering a torsional pendulum, where Io is the moment of inertia of the disk alone, Ic is the moment of inertia of the reference ring, and k is the torsion constant, the periods of oscillation can be determined using the formula for the period T of a torsion pendulum:

T = 2π√(I/k)

1. The period To for the disk alone is given by:

To = 2π√(Io/k)

2. When the reference ring is added to the disk, the total moment of inertia becomes I' = Io + Ic. The period T1 for the combined system is:

T1 = 2π√((Io + Ic)/k)

3. To express Io in terms of To, T1, and Ic, we take the ratio of T1 to T0:

(T1/To)2 = (Io + Ic)/Io

By rearranging and solving for Io, we find:

Io = Ic/((T1/To)2 - 1)