# Rotation and translation of a sphere in different directions

Consider a sphere of radius $$R$$ on a rough surface. Let it be rotating with angular velocity $$\vec{\omega}$$ and let it be moving with velocity $$\vec{v}$$. Then what happens after a very long time? What is the magnitude and direction of both its velocity vector and angular velocity vector after a long time?

EDIT: Let coefficient of static friction be $$\mu$$.

• Coefficient of friction can be assumed to be $\mu$? Commented May 26, 2020 at 11:53
• Nothing can be said in general. Commented May 26, 2020 at 11:59
• @HarishChandraRajpoot I don't think it should matter, but feel free to use the value as $\mu$ Commented May 26, 2020 at 12:52
• @Doubtnut something is got to happen, right? Are you saying that the same initial configuration can lead to different results? Commented May 26, 2020 at 12:54
• The sphere will execute pure rolling but its direction of motion can't be predicted. Commented May 26, 2020 at 12:56

Let the net frictional impulse by the ground be $$\vec{f}$$. This is until the sphere starts pure rolling.

Now the force $$\vec{f}$$ causes a change in the linear velocity of the sphere. The new velocity will be $$\vec{v_1} = \vec{v} + \frac{\vec{f}}{m}$$

Now let $$\vec{r}$$ be the vector which points from the center to the point of contact. Then our new $$\omega_1$$ will be $$\vec{\omega_1} = \vec{\omega} + \frac{\vec{f} \times \vec{r}}{I}$$

where $$I$$ is the moment of inertia of the sphere. In the end the point of contact should be at rest. So $$\vec{\omega_1} \times \vec{r} + \vec{v_1} = 0$$ $$\implies \ \vec{\omega} \times \vec{r} - \frac{\vec{f} \cdot R^2}{I} + \vec{v} + \frac{\vec{f}}{m} = 0$$

From this we can get $$\vec{f}$$, from which we can get $$\vec{w_1}$$ and $$\vec{v_1}$$.

Note: technically the second line should have been written as $$\Delta \vec{L} = \vec{f} \times \vec{r}$$, but the inertia is the same for a sphere along any axis going through its center, so we can get away with it.

You have to find the velocity of the bottom-most point wrt ground (v-Rω). The frictional force will continue to act against it until velocity of the point becomes zero. This can happen as the ball stops moving and rolling at the same time (this case needs a specific set of initial values) or starts pure rolling (happens in most of the cases).