Angular Momentum and ball velocity

[zeromaxxx]

Homework Statement

A billiard ball (solid uniform sphere) is struck so that it has initial velocity v₀, but no initial spin. As it slides along the table, friction causes it to spin, until eventually it is rolling without slipping. Find its speed when it begins to roll without slipping. Show that your answer does not require that the force of friction be constant (e.g., equal to μN).

Homework Equations

I = 2/5(mr²)

F_k = umg

w = ugt/r

T = F x r

T = I x (angular accel)

F=ma

V = -ugt

The Attempt at a Solution

Considering a constant force:
v = v₀ - ugt

rw = 5/2(ugt)

Equating both:

v = rw, t = 2/7 (v₀/ug)

w = 5/7 (v₀/r)

v = wr = 5/7 v₀

How do I show that it does not require that the force of friction be constant to get the same result?

 

[anathema]

To show that the result does not require a constant force of friction, we can use the equations for rotational motion and the definition of friction to derive the same result.

From the given information, we know that the billiard ball starts with an initial velocity v0 and no initial spin. As it slides along the table, friction causes it to spin until it reaches a state of rolling without slipping.

Using the equations for rotational motion, we can find the angular velocity of the ball at the moment it starts rolling without slipping.

First, we know that the moment of inertia (I) for a solid uniform sphere is equal to 2/5(mr^2).

Next, we can use the definition of friction (Fk = μmg) to find the torque (T) acting on the ball.

T = Fk x r = μmg x r

We can then use the equation T = I x α (where α is the angular acceleration) to find the angular acceleration of the ball.

α = T/I = μmg x r / (2/5mr^2) = (5/2)μg/r

Finally, we can use the equation w = ω0 + αt (where ω0 is the initial angular velocity) to find the angular velocity of the ball at the moment it starts rolling without slipping.

w = ω0 + αt = 0 + (5/2)μg/r x t = (5/2)μgt/r

We know that at the moment the ball starts rolling without slipping, the linear velocity (v) and angular velocity (w) are related by the equation v = rw.

Substituting the value of w we found above, we get v = (5/2)μgt.

This result is the same as the one we found earlier using the constant force of friction. However, the above derivation did not require the assumption of a constant force of friction. Therefore, we have shown that the result does not depend on the force of friction being constant.