Bug on a pivoted ring: rotational velocity

In summary, the problem discusses a ring of mass M and radius R that is pivoted to a frictionless table. A bug of mass m walks around the ring with speed v, starting at the pivot. The question is what is the rotational velocity of the ring when the bug is halfway around the pivot. The conversation also includes a discussion of which quantities are conserved, whether energy is conserved, and whether the bug still has a velocity v when it reaches the top of the ring. The equations -mv = m(v+V) + Mv and -mRv = -mR(v+V) + M(R^2)(omega) are used to solve for the angular velocity, but the answer does not match the
  • #1
Sam_Goldberg
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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.
 
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  • #2
Anyone?
 
  • #3
Sorry for the late reply but I was looking for it myself just now and happened to find it!

It's problem 1173 in the Lim Classical Mechanics problems and solutions manual.
 

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Related to Bug on a pivoted ring: rotational velocity

1. What is a pivoted ring in relation to rotational velocity?

A pivoted ring is a circular object that is attached to a pivot point and can rotate freely around that point. In the context of rotational velocity, it is often used as a model for understanding rotational motion and the effects of forces on rotating objects.

2. How does the bug on a pivoted ring experiment demonstrate rotational velocity?

In the bug on a pivoted ring experiment, a small bug is placed on a rotating ring and the ring is then spun at a constant speed. As the ring rotates, the bug experiences a centripetal force that causes it to move in a circular path. The speed at which the bug moves around the ring is known as the rotational velocity, and it can be measured and analyzed to understand the relationship between forces and rotational motion.

3. What factors affect the rotational velocity of the bug on a pivoted ring?

The rotational velocity of the bug on a pivoted ring is affected by the speed at which the ring is spun, the mass of the bug, and the distance between the bug and the pivot point. Other factors such as air resistance and friction may also play a role in the observed rotational velocity.

4. How is rotational velocity different from linear velocity?

Rotational velocity refers to the speed at which an object is rotating around an axis, while linear velocity refers to the speed at which an object is moving in a straight line. In other words, rotational velocity involves circular motion, while linear velocity involves straight-line motion.

5. What real-world applications use the concept of rotational velocity?

The concept of rotational velocity is used in many real-world applications, such as understanding the motion of planets and other celestial bodies, predicting the behavior of spinning objects like tops and gyroscopes, and designing machinery and vehicles that involve rotating parts, such as engines and turbines.

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