There is a hoop of mass m and a block of mass m is attached to it.The hoop purely rolls and initially the ball is at the bottom.Please describe the subsequent motion and the situation in which the hoop starts rising upwards???

To understand the motion of the hoop and the block attached to it, we need to consider the forces and torques acting on the system. When the hoop rolls without slipping, it experiences both translational and rotational motion. Let’s break down the scenario step by step.

Initial Setup

Imagine a hoop with mass "m" resting on a flat surface. Attached to the hoop is a block of the same mass "m," positioned at the bottom of the hoop. Initially, the system is at rest. The key point here is that the hoop can roll, which means it can rotate about its center of mass while also moving linearly.

Forces Acting on the System

When the block is at the bottom of the hoop, gravity acts on both the hoop and the block. The gravitational force on the block creates a torque about the center of the hoop. This torque will cause the hoop to start rolling. The forces involved include:

• Gravitational Force: The weight of the block (mg) acts downward.
• Normal Force: The ground exerts an upward normal force on the hoop.
• Frictional Force: Static friction between the hoop and the ground prevents slipping and allows rolling.

Subsequent Motion

As the hoop begins to roll, the block will also start to move due to the gravitational force acting on it. The motion can be described as follows:

• The hoop rolls forward, and the block moves downward due to gravity.
• As the block descends, it will exert a force on the hoop, causing it to rotate.
• The center of mass of the hoop-block system will accelerate as the hoop rolls.

When Does the Hoop Start Rising?

The hoop will begin to rise when the block reaches a certain height. This occurs when the gravitational force acting on the block generates enough torque to lift the hoop. Specifically, if the block moves upward to a point where its weight can create a torque that exceeds the downward force acting on the hoop, the hoop will start to rise. This can happen under the following conditions:

• The block must ascend to a height where the center of mass of the system shifts sufficiently.
• The hoop's rolling motion must be fast enough to allow the block to gain vertical momentum.

Example Scenario

Consider a situation where the hoop rolls down a slope. As the block moves up the incline of the hoop, it gains height. If the incline is steep enough, the gravitational force acting on the block can create a torque that lifts the hoop off the ground. The hoop will rise until the block reaches a new equilibrium position, where the forces balance out.

Conclusion

In summary, the motion of the hoop and block is a fascinating interplay of forces and torques. The hoop rolls due to the gravitational force acting on the block, and it will rise when the block's weight generates sufficient torque to lift the hoop. Understanding these dynamics helps in grasping the principles of rotational motion and the effects of gravity on different masses.