Question

A ball of mass m and radius R rotates with an angular velocity ω_0 about a horizontal axis. If it is placed on a horizontal plane, the coefficient of friction being µ, how far can the ball travel before it is engaged in a pure rolling motion ?

Answer 1

Answer on Question #61803 - Physics - Mechanics | Relativity

Question:

A ball of mass $m$ and radius $R$ rotates with an angular velocity $\omega_0$ about a horizontal axis. If it is placed on a horizontal plane, the coefficient of friction being $\mu$, how far can the ball travel before it is engaged in a pure rolling motion?

Solution:

$$\vec{F} = m \vec{a};$$ $$\left\{ \begin{array}{l} F = m a, \\ F = \mu m g; \end{array} \right.$$

Now we will find angular acceleration:

$$\vec{M} = I \vec{\beta} = [\vec{R}, \vec{F}];$$ $$M = I \beta = F R;$$ $$I \beta = F R;$$ $$\frac{2}{5} m R^2 \beta = \mu m g R \Rightarrow \beta = \frac{5 \mu g}{2 R};$$

Now we will find angular velocity:

$$m a = \mu m g \Rightarrow a = \mu g \Rightarrow \frac{V}{t} = \mu g \Rightarrow \frac{\omega R}{t} = \mu g \Rightarrow \omega = \frac{\mu g t}{R};$$

Now we can find time of motion without pure rolling:

$$\omega = \omega_0 - \beta t \Rightarrow \frac{\mu g t}{R} = \omega_0 - \frac{5 \mu g}{2 R} t \Rightarrow t = \frac{2 R \omega_0}{7 \mu g};$$

And finally, we can find how far the ball can travel before it is engaged in a pure rolling motion:

$$S = \frac{a t^2}{2} = \frac{4 \mu g R^2 \omega_0^2}{2 \cdot 49 \mu^2 g^2} = \frac{2 R^2 \omega_0^2}{49 \mu g};$$

Answer:

$$S = \frac{2 R^2 \omega_0^2}{49 \mu g};$$

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