Uniform Solid Sphere on Rough Surface

When a uniform solid sphere of unit radius is given a translational velocity V₁ and angular velocity wo and placed on a rough surface, it will eventually come to rest completely after moving some distance. This can be explained based on the principles of rotational and translational motion.

1. Initial Motion
- The sphere has an initial translational velocity V₁, which means it is moving in a straight line.
- The sphere also has an initial angular velocity wo, which means it is rotating about its center.
- These two motions combine to give the sphere a complex motion.

2. Rolling Motion
- Due to the rough surface, there is a frictional force acting on the sphere in the opposite direction of its motion.
- This frictional force exerts a torque on the sphere, causing it to decelerate in both translation and rotation.
- As the sphere rolls, the point of contact with the surface has zero velocity relative to the ground.

3. Work Done by Frictional Force
- The frictional force does negative work on the sphere, as it opposes the motion.
- This work is converted into rotational kinetic energy and translational kinetic energy.
- As the sphere decelerates, the work done by friction decreases until it eventually comes to rest.

4. Conversion of Energy
- As the sphere slows down, its rotational kinetic energy decreases.
- This energy is converted into translational kinetic energy, causing the sphere to move forward.
- Eventually, all the rotational kinetic energy is converted into translational kinetic energy and the sphere comes to rest.

5. Distance Traveled
- The distance traveled by the sphere before it comes to rest can be calculated using the equations of motion.
- The deceleration of the sphere can be determined by considering the torque due to friction and the moment of inertia of the sphere.
- Using the equations of rotational motion and translational motion, the distance traveled can be calculated.

In conclusion, when a uniform solid sphere with initial translational and angular velocities is placed on a rough surface, it will eventually come to rest completely after moving some distance. This is due to the frictional force exerting a torque on the sphere, causing it to decelerate in both translation and rotation. The work done by the frictional force converts the sphere's kinetic energy into translational and rotational kinetic energy until all of it is dissipated and the sphere comes to rest. The exact distance traveled can be calculated using the equations of motion.