Angular Momentum Homework: Ratio of ω0 to V0 for Zeroed Values

[Karol]

Homework Statement

A finger presses a ball and throws it away with angular velocity ω₀ and velocity V₀ like in the picture. the coefficient of friction is μ. what is the ratio ω₀ to V₀ so that they both are zeroed at the same time

Homework Equations

$$I_C=\frac {2}{5}mr^2$$
$$Torque=\dot{L}$$
From kinematics:
$$V^2=V_0^2-2ax,\quad \omega^2=\omega_0^2-2\alpha\theta$$

The Attempt at a Solution

The friction decelerates the ball:
$$mg\mu r=\frac{2}{5}mr^2\cdot\alpha\Rightarrow\alpha=\frac{5\mu g}{2r}$$
from kinematics:
$$0=\omega_0^2-2\alpha\theta\Rightarrow\theta=\frac{\omega_o^2 r}{5g\mu}$$
The distance travelled:
$$x=\theta r=\frac{\omega_o^2 r^2}{5g\mu}$$
I insert this distance into the kinematics formula:
$$0=V_0^2-2g\mu\frac{\omega_o^2 r^2}{5g\mu}\Rightarrow V_0=\sqrt{\frac{2}{5}}\omega_0 r$$
And the answer should be:
$$V_0=\frac{2}{5}\omega_0 r$$

 

[TSny]

The distance travelled:
$x=\theta r$

Is this true if the ball slips?

Try starting with kinematic equations that involve time, since you know that ω and v come to zero at the same time.