- #1

Sam_Goldberg

- 46

- 1

## Homework Statement

Hi guys, this question is from Kleppner and Kolenkow, problem 6.3. A ring of mass M and radius R lies on its side on a frictionless table. It is pivoted to the table at its rim. A bug of mass m walks around the ring with speed v, starting at the pivot. What is the rotational velocity of the ring when the bug is halfway around the pivot?

## Homework Equations

## The Attempt at a Solution

We will take the bug to be moving to the left at speed v initially. I would like to ask you guys a few questions. First: which quantities are conserved? It seems obvious that total linear momentum in the x-direction is conserved. I would also argue that angular momentum is conserved, since the only external forces on the system are gravity and the normal force, but these create a zero net torque. I'm not sure whether energy is conserved. For the bug to walk around the ring, there needs to be friction between it and the ring; however, is this friction dissipative?

I have a second question: when the bug gets to the top, does it still have a velocity v (directed the the right) with respect to the ring, or not?

If I assume that only momentum and angular momentum are conserved, and that the bug still has speed v with respect to the ring, I get these equations:

-mv = m(v + V) + Mv

-mRv = -mR(v + V) + M(R^2)(omega)

where V is the velocity of the ring and omega is the angular velocity. I solve these equations for omega, and I don't get the answer in the book.

Where did I go wrong? Thanks in advance for all your help.