Rotational Motion: Meaning of ∫v(t)dt - ∫Rw(t)dt

[gsimo1234]

This is merely a simple, but conceptual, problem. Say we have a cue ball of mass M and Radius R rolling without slipping on the pool table. What is the the meaning of the ∫v(t)dt - ∫Rw(t)dt where w(t) is the angular speed of the pool ball.

My guess is that this represents the length the ball has moved and the amount of radians that the ball has spun through. Am I correct?

 

[kuruman]

You are correct about the first integral but not the second. Do dimensional analysis and you will see why.

 

[kerimek]

Your guess is partially correct. The integral of v(t)dt represents the displacement of the ball, or the distance it has moved along its path. This is because v(t) is the velocity of the ball at any given time, and integrating it over time gives us the change in position.

The integral of Rw(t)dt, on the other hand, represents the angular displacement of the ball. This is because w(t) is the angular speed of the ball, and integrating it over time gives us the change in the angle through which the ball has rotated.

Therefore, the difference between these two integrals, ∫v(t)dt - ∫Rw(t)dt, represents the net displacement of the ball, taking into account both its linear and angular motion. In other words, it represents the total distance and direction that the ball has moved on the pool table.

It's also worth noting that since the ball is rolling without slipping, the relationship between its linear and angular speed is given by v = Rw, where v is the linear speed, R is the radius, and w is the angular speed. This means that we can rewrite the integral of Rw(t)dt as ∫v(t)/R dt, which further emphasizes the connection between the two integrals.

In summary, the integral of v(t)dt - ∫Rw(t)dt represents the net displacement of a rolling ball on a pool table, taking into account both its linear and angular motion. It is a useful concept in understanding the overall motion of rotating objects.