Rotational Dynamics Homework: 2 Rings M & 4M

[Mandeep Deka]

Homework Statement

Two rings of masses 'm' (smaller one) and '4m' (bigger one) are placed in the set up as shown in the diagram. The smaller ring is hung through a horizontal frictionless thread. The bigger ring is joined to the smaller ring in such a way that it can rotate freely about the point of suspension. The planes of both the rings are perpendicular to the plane of the paper. The bigger ring has a radius 'R'. After the bigger ring is released from the initial position as shown in the figure, find the angular velocity of the bigger ring when its plane becomes vertical i.e the planes of both the rings become vertical.

Homework Equations

The Attempt at a Solution

First of all, when the rings reach the final position, we can calculate the relation between the velocities of the center of masses of both the rings (their velocity vectors will be horizontal and since there is no ext. force in the horizontal direction we can conserve the linear momentum). Now when the bigger ring reaches the final position the decrease in the potential energy of the system is, (4m)gR, which will be converted into kinetic energies of the bodies. But i have a doubt; the K.Es of the smaller ring will be only its linear K.E, but for the bigger ring, what will be the K.E? Will it just the linear K.E of the COM of the ring at that instant? or it will be the rotational energy about the point of suspension? or both?

The answer to the question is (20g/17R)^(1/2), but in no ways i am getting to it.
Please help!

 

[zorro]

Good question. I think that there will be rotational + tranlational kinetic energy of the bigger ring.
Tranlational kinetic energy of to COM if not there, will violate the conservation of angular momentum.

 

[Mandeep Deka]

First, About which point are you thinking to conserve angular momentum, because there will always be an external torque at the COM of the bigger ring, which will rotate it about the point of suspension?
Second, why will you consider the translational K.E of the ring, if you already consider the whole K.E of the body (in the form of the rotational KE) about the point of suspension?

Please try to find out the answer, it matches neither ways!

 

[Mandeep Deka]

Is there anyone??
Plz. help, i am having a terrible time, understanding the problem!

 

[rwisz]

I would approach this problem by first identifying the relevant equations and principles in rotational dynamics. In this case, we have two rings of different masses, connected in a system where one is rotating freely about a fixed point. We can use the principle of conservation of angular momentum to analyze this system.

We know that when the bigger ring reaches the final position, the decrease in potential energy will be converted into kinetic energy. The smaller ring will only have linear kinetic energy, but for the bigger ring, it will have both linear and rotational kinetic energy. This is because the bigger ring is rotating about its center of mass as well as around the point of suspension.

To find the final angular velocity of the bigger ring, we can use the equation for conservation of angular momentum:

I1ω1 = I2ω2

Where I is the moment of inertia and ω is the angular velocity. We can also use the equation for the moment of inertia of a ring:

I = MR^2

Using these equations, we can solve for the final angular velocity of the bigger ring when its plane becomes vertical. It is important to note that the final angular velocity will be a combination of the linear and rotational velocities, so we cannot simply use the equation for linear kinetic energy.

I would recommend reviewing the equations and principles of rotational dynamics to better understand this problem and arrive at the correct solution. It may also be helpful to draw a free body diagram and carefully consider the forces and motions involved in the system. Good luck with your homework!