A hoop of radius r and mass m rotating with an angular velocity 0 ? is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

To understand the motion of a hoop placed on a rough surface, we need to consider the concepts of rotational motion and friction. When the hoop is initially at rest with respect to its center, but rotating, it will experience friction when placed on a rough surface. This frictional force will act to prevent slipping and will eventually lead to a condition where the hoop rolls without slipping. Let's break this down step by step.

Understanding the Forces at Play

When the hoop is rotating with an angular velocity (ω) and is placed on a rough surface, the frictional force between the hoop and the surface plays a crucial role. The friction acts in the direction opposite to the motion of the center of mass of the hoop. Since the center of the hoop is initially at rest, the friction will exert a force that accelerates the center of the hoop.

Friction and Acceleration

The frictional force can be expressed as:

• Frictional Force (f) = μN, where μ is the coefficient of friction and N is the normal force (equal to mg for a horizontal surface).

This frictional force will cause a linear acceleration (a) of the center of the hoop. According to Newton's second law, we have:

• F = ma

Thus, the acceleration of the center of the hoop can be given by:

• a = f/m = μg

Relating Linear and Angular Motion

As the hoop rolls without slipping, there is a relationship between the linear velocity (v) of the center of the hoop and the angular velocity (ω) of the hoop:

• v = rω

Initially, the hoop has an angular velocity ω but no linear velocity. As it rolls, the frictional force will cause the center of the hoop to gain velocity while simultaneously affecting its angular velocity.

Final Velocity Calculation

When the hoop ceases to slip, the condition for rolling without slipping is satisfied. At this point, the linear velocity of the center of the hoop (v) will equal the product of its radius (r) and its angular velocity (ω) at that moment:

• v = rω', where ω' is the new angular velocity after rolling without slipping.

Using the conservation of angular momentum, we can relate the initial angular momentum to the final state. The initial angular momentum (L_initial) of the hoop is:

• L_initial = Iω, where I is the moment of inertia of the hoop, given by I = mr².

Thus, the initial angular momentum is:

• L_initial = mr²ω.

When the hoop rolls without slipping, the final angular momentum (L_final) is:

• L_final = Iω' = mr²ω'.

Setting these equal gives us:

• mr²ω = mr²ω'.

From this, we can see that the angular velocity will adjust to maintain the relationship between linear and angular motion. The final velocity of the center of the hoop when it ceases to slip can be expressed as:

• v = rω'.

Conclusion

In summary, when the hoop ceases to slip, the velocity of the center of the hoop will be equal to the product of its radius and the angular velocity at that moment. If we assume that the hoop starts with an angular velocity ω and friction is sufficient to prevent slipping, the final velocity of the center of the hoop will be:

• v = rω.

This means that the center of the hoop will move forward with a velocity proportional to its radius and its initial angular velocity, effectively transitioning from pure rotation to rolling motion without slipping.