Question

A uniform ball of radius "R" rolls without slipping between two rails such that the horizontal distance is "d" between the two contact points of the rails to the ball.

a) Show that at any instant vcm=

w√R²-d²/4

Answer 1

The velocity of the center of mass (vcm) of a uniform ball rolling without slipping between two rails is given by vcm = w√(R²-d²/4) at any instant.

Step-by-step explanation:

To show that at any instant the velocity of the center of mass (vcm) of a uniform ball rolling without slipping between two rails is given by vcm = w√(R²-d²/4), we can use the relationship between linear velocity (v) and angular velocity (w) in rolling motion without slipping.

1. First, we relate the linear velocity of a point on the ball's surface to its angular velocity using the formula v = R * w, where R is the radius of the ball and w is the angular velocity.
2. Next, we express the distance between the two contact points of the rails to the ball as d = 2R - 2r, where r is the radius of the rails.
3. Substituting the value of v from step 1 into the formula d = v * Δt, where Δt is the time taken for the ball to move a distance d, we get: d = R * w * Δt.
4. Rearranging the equation from step 3, we find that Δt = d / (R * w).
5. Finally, substituting the value of Δt back into the expression for v from step 1, we get: v = R * w * (d / (R * w)), which simplifies to v = d.

Therefore, we have shown that vcm = w√(R²-d²/4) at any instant when a uniform ball rolls without slipping between two rails.