A bowling ball of mass M and radius R is thrown along a level surface so that initially ( t= 0) it slides with a linear speed but does not rotate, see the figure. As it slides, it begins to spin, and eventually rolls without slipping.

How far has the ball moved down the lane when it starts rolling without slipping?

What are its final linear speed?

What are its final rotational speed?

Thanks:]

You need to specify kinetic friction coefficient and the initial velocity.

it says to answer it in terms of those coefficients

like, it's asking for formulas...

Initial w, angular velocity, is zero.

now while sliding
torque = I alpha
mu m g r = (2/5) m r^2 alpha
alpha = (5/2) mu g / r
and
F = m a
mu m g = m a
a = mu g
w = alpha t
v = Vi - a t
when does w r equal v (no slip)
r alpha t = v = Vi - a t
(5/2) mu g t = Vi - mu g t
(7/2)mu g t = Vi
so
t = (2/7)Vi/(mu g)
that is the time sliding
work back up to get distance and final v

Check that with energy argument:
frictional work done = mu m g d

Initial Ke - frictional work = Final Ke
(1/2) m Vi^2 - mu m g d = (1/2) m v^2 +(1/2) (2/5) m r^2 (v^2/r^2)

Vi^2 - 2 mu g d = (7/5)v^2

To determine the distance the ball has moved down the lane before it starts rolling without slipping, we need to consider the conditions for rolling without slipping. When the ball starts rolling without slipping, the linear speed of the ball equals the tangential speed at the point of contact with the surface.

To find the distance, we can use a combination of the equations of linear and rotational motion. Let's break down the steps:

1. Identify the condition for rolling without slipping:
When the ball starts rolling without slipping, the linear speed of the ball (V) is equal to the tangential speed (v) at the point of contact.

2. Determine the initial linear speed:
The initial linear speed of the ball (V₀) is given in the problem statement.

3. Find the initial rotational speed:
The initial rotational speed (ω₀) can be calculated using the formula: ω₀ = V₀ / R, where R is the radius of the ball.

4. Determine the time taken for the ball to start rolling without slipping:
The time taken for the ball to start rolling without slipping can be calculated using the equation: t = V₀ / (α×R), where α is the rotational acceleration.

5. Calculate the distance moved down the lane:
The distance moved down the lane (d) can be calculated using the equation: d = V₀×t + (1/2)×α×R×t².

Now let's calculate the final linear speed and rotational speed:

6. Calculate the final linear speed:
The final linear speed is given by V = v = V₀ as mentioned earlier.

7. Calculate the final rotational speed:
The final rotational speed (ω) can be calculated by dividing the final linear speed by the radius of the ball, as ω = V / R.

By following these steps, you should be able to solve the problem and find the distance moved down the lane, as well as the final linear and rotational speeds of the ball.