Rigid body/perfectly plastic collision question

[fluidistic]

Homework Statement

Today a friend of mine came to my apartment and we discussed a problem. I realized something that surprises me A LOT.
Consider the following situation. There are 2 disks of radius R and mass M on a frictionless table. One doesn't move while the other suffers a translation with speed v and will hit the quiet disk such that its lowest point will touch the upper point of the other disk. When it hits it, the two disks remain attached forming a new rigid body.
Calculate the angular velocity of the rigid body.
I think one gets the answer using the conservation of angular momentum or something like that. The result is $$\frac{v}{6R}$$. I didn't realized it and thought that the kinetic energy would be conserved and I got a result of $$\omega=\frac{v}{\sqrt{6}R}$$. As $$\omega=\frac{v}{\sqrt{6}R} > \frac{v}{6R}$$ it means the system loses kinetic energy by a factor $$\frac{1}{\sqrt 6}$$. Why such a number? Is that true with any plastic collision?
And another question : How does this energy converts itself into heat? There's no friction nor elastic deformation due to compression since we're talking about rigid bodies. I don't see any way to convert the energy. Hence I'm missing something... can you help me answering some (or all) these questions ?
Thanks.

 

[jbsteele]

Homework EquationsConservation of Angular Momentum: L = IωConservation of Kinetic Energy: K = 1/2mv^2 + 1/2Iω^2The Attempt at a SolutionThe angular velocity of the rigid body can be found using conservation of angular momentum. The initial angular momentum of the two disks is 0, since they are not rotating and their moment of inertia is 0. Therefore, the final angular momentum must also be 0. Since the angular momentum of a rigid body is equal to its moment of inertia multiplied by its angular velocity, the angular velocity must then be 0. The kinetic energy of the system can be found using conservation of kinetic energy. The initial kinetic energy of the two disks is 0, since they are not moving. Therefore, the final kinetic energy must also be 0. Since the kinetic energy of a system is equal to 1/2mv^2 + 1/2Iω^2, the angular velocity must then be 0. This means that the energy is not converted into heat, since there is no heat generated in a plastic collision. Instead, the energy is converted into potential energy due to the deformation of the disks.

 

[thomate1]

I can provide some insight into this problem and the surprising results you have found. First, let's address the issue of energy conservation. In a perfectly plastic collision, kinetic energy is not conserved because some of the energy is converted into other forms, such as heat or sound. This is due to the fact that the colliding objects deform and stick together, rather than bouncing off each other like in an elastic collision.

Now, let's look at the conservation of angular momentum. This principle states that the total angular momentum of a system remains constant unless acted upon by an external torque. In this case, the two disks form a new rigid body, so the angular momentum of the system must remain the same before and after the collision. This is why you were able to use the conservation of angular momentum to calculate the resulting angular velocity.

As for the specific value of \frac{v}{6R}, this is a result of the specific geometry and conditions of the problem. It is not a universal value for all perfectly plastic collisions. It may vary depending on the masses, radii, and velocities of the objects involved.

To address your question about the conversion of energy into heat, it is important to note that even though the disks are considered rigid bodies, they still have some degree of flexibility and can deform slightly during the collision. This deformation causes some energy to be converted into heat, which is then dissipated into the surrounding environment.

I hope this helps clarify some of your questions and provides a better understanding of the principles at play in this problem. As always, it is important to carefully consider the assumptions and conditions of a problem before drawing conclusions.