- #1

Order

- 97

- 3

## Homework Statement

A solid disk of mass

*M*and radius

*R*is on a vertical shaft. The shaft is attached to a coil spring which exerts a linear restoring torque of magnitude [tex]C\theta,[/tex] where

*theta*is the angle measured from the static equilibrium position and

*C*is constant. Neglect the mass of the shaft and the spring, and assume the bearings to be frictionless. (Please tell me if you want me to attach a figure as well. That is a lot of work but can be arranged.)

a. Show that the disc can undergo simple harmonic motion, and find the frequency of the motion.

b. Suppose that the disc is moving according to [tex]\theta=\theta_{0}\sin(\omega t),[/tex] where

*omega*is the frequency found in part a. At time [tex]t_{1}=\pi / \omega ,[/tex] a ring of sticky putty of mass

*M*and radius

*R*is dropped concentrically on the disk. Find:

(1) The new frequency of the motion

(2) The new amplitude of the motion

## Homework Equations

[tex]I_{disc}=\frac{MR^{2}}{2}[/tex]

[tex]I_{ring}=MR^{2}[/tex]

## The Attempt at a Solution

a. The initial moment of inertia is [tex]I_{0}=\frac{MR^{2}}{2}[/tex] and the torque is defined as [tex]\tau = I \ddot{\theta}.[/tex] In this case the torque is [tex]\tau=-C\theta,[/tex] which leads to the equation [tex]\ddot{\theta}=-\frac{2C}{MR^{2}}\theta,[/tex] with the solution [tex]\theta=\theta_{0}\sin (\omega t),[/tex] where [tex]\omega=\sqrt{\frac{2C}{MR^{2}}}.[/tex]

b. Differentiating the equation leads to [tex]\dot{\theta}=\theta_{0}\omega\cos (\omega t),[/tex] so [tex]\dot{\theta}(\pi / \omega)=-\theta_{0} \omega,[/tex] so the disc has a certain velocity. Calculating the new frequency is easy: [tex]\omega_{new}=\sqrt{\frac{C}{\frac{MR^{2}}{2}+MR^{2}}}=\sqrt{\frac{2C}{3MR^{2}}}[/tex] Now since the angular momentum is conserved [tex]\theta_{0}\omega=\theta_{new}\omega_{new},[/tex] so [tex]\theta_{new}=\theta_{0}\sqrt{3}.[/tex]

Now can this be correct? The energy is defined as [tex]\frac{I\omega^{2}}{2}.[/tex] So in this case [tex]E_{i}=\frac{I_{0}\omega^{2}}{2}=C[/tex] and [tex]E_{f}=\frac{(I_{0}+I_{ring})\omega_{new}^{2}}{2}=C[/tex] so the energy is conserved. I don't understand this because there is a collision and some energy should be lost! Does this mean that my calculation is incorrect. And if so, where did I go wrong?