Conservation of Momentum vs Energy

[Student149]

Homework Statement

Given a horizontal rod (parallel to x axis) and two disks attached near its ends symmetrically. The rods and disks are perfectly balanced i.e. the disks have same mass and density. The system's center of mass is exactly at geometric center of the rod (since its balanced). The disks are attached at their center with the rod and are free to rotate about the pivot point parallel to x axis. The complete setup is at rest in space w.r.t. to the external observer.

There are two point masses (each with mass = m₀) that are traveling towards the rod parallel to y-axis at a velocity v₀. They stick to the setup after the collision.

Now consider two cases:

Case 1: The 2 point masses hit the rod at equal distance from its center of mass. They stick to the rod after collision. The combined system has a linear velocity = v₁ along y axis.

Case 2: The 2 point masses hit the disk's edges equidistant from the center of mass of the whole system. They stick to the disk edge after collision. The point of contact is such that one disk rotate in opposite direction (one clockwise another anticlockwise) with equal velocity.The system thus, still has net 0 angular momentum. The combined system has a linear velocity = v₂ along y axis.

Would the velocity of the system be same in both cases along y-axis as seen by the external observer?

Homework Equations

The Attempt at a Solution

My understanding is as some of the energy is used in disk rotation in case 2 when the to point masses collide with the system, where as in case 1 all energy is used in head on collision in y axis. Thus, the velocity of the system in case 2 would be lesser than the velocity of the system in case 1.

But, if conservation of momentum is taken literally, no matter where the 2 point masses hit the system (such that they are equidistant from center of mass of the system), the velocity of the system along y-axis would be same. Thus, case 1 and case 2 both would have same velocity along y axis.

They seem to be contradictory. Which line of argument is correct and why?

 

[Orodruin]

They stick to the setup after the collision.

This means that the collision is inelastic, i.e., some of the energy is converted into internal energy (e.g., in the form of heat). It is therefore not possible to use conservation of kinetic energy.

 

[Student149]

This means that the collision is inelastic, i.e., some of the energy is converted into internal energy (e.g., in the form of heat). It is therefore not possible to use conservation of kinetic energy.

Thank you. So as I understand in case of inelastic collisions as above, the linear velocity should be same (as in the above setup) along y axis, no matter where the point masses make contact with the setup (either disk or rod) ?

And the above would also be true if the setup was making some angle w.r.t. the x-axis to begin with (the angle is w.r.t. the center of mass of the rod/setup) ?

 

[Orodruin]

Just from conservation of total momentum (assuming no external forces), the centre of mass of the objects always travels at the same velocity.