# Perfectly inelastic collision of two moving and rotating disks

• mattlfang
In summary, two moving and rotating uniformly weighted disks collide in a perfectly inelastic collision. They are rotating in opposite directions and at the moment of collision, the angles between their velocity and the line connecting their centers are 45 degrees. The velocities are therefore at right angles. We have the angular velocity, velocities, radii, and masses of the disks. Using the center of mass as the reference, we can ignore linear momentum and focus on angular momentum. After collision, the disks will continue to rotate about their combined center of mass, similar to two intermeshing gears.
mattlfang
Homework Statement
Perfectly inelastic collision of two moving and rotating disk
Relevant Equations
angular momentum, momentum
two moving and rotating, uniformly weighted disks perfectly inelastic collide. The disks are rotating in opposite directions (see the diagram) At the moment of their collision, the angles between their velocity and the line connecting their centers are 45 degrees. The velocities are therefore in right angle. We know their angular velocity, their velocities, their radii and their masses. Find the angular velocity after collision.

My thought. Let the center of the mass of the entire system be our reference. We can then ignore the linear momentum and only work on the angular momentum.
The angular momentum of wheel 1 w.r.t center of mass of the whole system is

I1 ω1+ m1 v'1, where v'1 is the vertical relative velocity with respect to the center of the mass of the system,

similarly compute angular momentum of wheel 2 w.r.t center of mass of the whole system.

But I got stuck on computing v'1

Can someone first comment if my above approach is correct, also comment how to find v'1 ?

Last edited:
Using the CoM frame looks like the way to go. The amalgamated discs are then spinning about their CoM after the collision.

The diagram has information not present in the description. It shows them rotating in opposite directions, and with the velocities at right angles. From the description alone, they could have been rotating the same way and/or moving anti parallel.

Lnewqban
haruspex said:
The diagram has information not present in the description. It shows them rotating in opposite directions, and with the velocities at right angles. From the description alone, they could have been rotating the same way and/or moving anti parallel.
they are rotating in opposite directions yes, but I did mention the velocities are at right angles, let me make that more clear.

mattlfang said:
Homework Statement:: Perfectly inelastic collision of two moving and rotating disk
Relevant Equations:: angular momentum, momentum

where v'1 is the vertical relative velocity
Do you mean the component of the relative velocity at right angles to the line of centres?

haruspex said:
Do you mean the component of the relative velocity at right angles to the line of centres?
yes, exactly, but it doesn't look easy to express it.

mattlfang said:
they are rotating in opposite directions yes, but I did mention the velocities are at right angles, let me make that more clear.
No, post #1 says the angles between their velocity and the line connecting their centers are 45 degrees. So the velocities could be at right angles, parallel, or antiparallel.

mattlfang said:
yes, exactly, but it doesn't look easy to express it.
If you find the velocity relative to the COM, it should be straightforward to take the component at that angle. Is it the first part of that you are unsure about?

haruspex said:
If you find the velocity relative to the COM, it should be straightforward to take the component at that angle. Is it the first part of that you are unsure about?

here is the diagram I drew. I assume wheel A has mass = 1, wheel B has mass = 3, velocity of A is (1,0), and velocity of B is (0,2). Then the velocity of the center of the mass of the system is (0.25, 1.25) vector OD. The velocity relative to the center of the mass of the system is vector AG (0.75, -1.25)

But I struggle to geometrically express its component orthogonal to OA using ##m_1, m_2, v_1, v_2##

separately, after two disks collide perfectly inelastically, do they both stop rotating with respect to their own centers, or do they keep rotating, sort of "rolling without slipping," similar to two intermeshing gears?

mattlfang said:
geometrically express its component orthogonal to OA
You have a component 0.25 in the X direction. What is the component of that normal to OA? Likewise for the Y component, and add them.
mattlfang said:
sort of "rolling without slipping," similar to two intermeshing gears?
I would say that is what is intended.

## 1. What is a perfectly inelastic collision?

A perfectly inelastic collision is a type of collision in which two objects stick together after impact and move as one object. This means that the kinetic energy of the system is not conserved and is converted into other forms of energy, such as heat or sound.

## 2. How is a perfectly inelastic collision different from an elastic collision?

In an elastic collision, the two objects bounce off each other and continue moving separately after impact. In a perfectly inelastic collision, the two objects stick together and move as one object after impact.

## 3. What factors affect the outcome of a perfectly inelastic collision?

The mass, velocity, and rotational motion of the objects involved in the collision can affect the outcome. Additionally, the angle of impact and the coefficient of restitution (a measure of the elasticity of the objects) can also play a role.

## 4. How is the momentum conserved in a perfectly inelastic collision?

In a perfectly inelastic collision, the total momentum of the system before and after the collision is conserved. This means that the total momentum of the two objects before impact is equal to the total momentum of the combined object after impact.

## 5. What are some real-life examples of perfectly inelastic collisions?

Some examples of perfectly inelastic collisions include a car crashing into a wall, a ball sticking to a wall after being thrown at it, or two trains colliding and sticking together. In these situations, the objects involved lose their individual identities and move as one object after impact.

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