A uniform wheel of radius R is made to rotate about its axis at an angular speed . The rotating wheel is then placed on a rough horizontal surface, keeping its axis horizontal. Due to friction at the point of contact, the wheel accelerates forward while its rotational speed decreases until pure rolling begins. Determine the linear speed of the wheel once it achieves pure rolling.

Step-by-step solution

Let the mass of the uniform wheel be m . A uniform wheel is considered as a uniform disc, so its moment of inertia about its central axis is I = 1 2 mR^2 . When the rotating wheel is placed on the rough horizontal surface, the frictional force acts at the point of contact. The normal reaction and gravitational force pass through the center of mass and are equal and opposite. Therefore, the net torque about any point on the ground along the line of motion is zero, and the angular momentum of the wheel about the poin