Please imagine you have a chassis (mass M) that is placed on rails (it can move only to one direction i.e axis y). On the chassis centre we have placed a rod that can rotate for half cycle (180 degs) with a high angular velocity and for the rest half cycle it rotates with a very small angular velocity. Also, the starting point for the rotating rod is when it is parallel to the rails (axis y). Due to the rotating rod the chassis will move on the rails. Question is: what is the relationship between the chassis's linear momentum and the rotating rod's angular momentum? Thank you!
The answer lies in Newtons THird law of motion. What is driving the rod rotation? Is it attached to the chassis? If so, Newton's third law is the answer.
The rod is driven by a 2-phased motor and it is attached on the chassis. What do you think is the chassis's velocity when the fast half-rotation is complete? Acccording to action-reaction it should be zero ... but is it?
Based on Newton's third law, for every action, there is an equal and opposite reaction. In this case, when the rod rotates with a high angular velocity, it exerts a force on the chassis, causing it to move. Simultaneously, the chassis exerts an equal and opposite force on the rod.
To determine the relationship between the chassis's linear momentum and the rotating rod's angular momentum, we need to consider the conservation of angular momentum.
Angular momentum is defined as the product of the moment of inertia and the angular velocity. In this scenario, the rotating rod has angular momentum due to its rotation. The chassis, on the other hand, has linear momentum due to its motion on the rails.
During the phase when the rod rotates with a high angular velocity, it transfers some of its angular momentum to the chassis in the form of linear momentum. This results in the chassis gaining linear momentum and moving along the rails.
However, during the phase when the rod rotates with a very small angular velocity, it transfers negligible angular momentum to the chassis. As a result, the chassis maintains its linear momentum during this phase.
Therefore, the relationship between the chassis's linear momentum and the rotating rod's angular momentum is that the chassis gains linear momentum when the rod rotates with a high angular velocity but retains its linear momentum when the rod rotates with a small angular velocity.