Solving for Rotational Velocity of a Ring and Bug System | Homework Question

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SUMMARY

The discussion revolves around calculating the rotational velocity of a ring with mass M and radius R, pivoted at its rim, as a bug of mass m walks around it. The user attempts to apply conservation of momentum and angular momentum but encounters discrepancies with the expected results. Key equations include linear momentum (p=mv) and angular momentum equations involving the bug's speed v, the ring's velocity V, and angular velocity omega. Clarification on the pivoting mechanism is also sought, indicating a need for deeper understanding of the system's dynamics.

PREREQUISITES
  • Understanding of conservation laws in physics, specifically momentum and angular momentum.
  • Familiarity with rotational dynamics and the concept of pivot points.
  • Basic knowledge of equations of motion for both linear and rotational systems.
  • Ability to manipulate and solve algebraic equations involving multiple variables.
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  • Study the principles of conservation of angular momentum in rotating systems.
  • Learn about the dynamics of pivoted bodies and their motion equations.
  • Explore examples of similar problems involving rotating rings and moving masses.
  • Review the effects of frictionless surfaces on motion and momentum conservation.
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Students studying physics, particularly those focusing on mechanics, as well as educators seeking to clarify concepts related to rotational motion and momentum conservation.

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1. Homework Statement

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 (a) is halfway around and (b) back at the pivot?

2. Homework Equations

p=mv

3. The Attempt at a Solution

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.
 

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"It is pivoted to the table at its rim."

What does this mean?
 
LawrenceC said:
"It is pivoted to the table at its rim."

What does this mean?

It is in attached picture
 

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