Angular Speed of a Hoop at the Bottom of a Ramp

Calculation Using Conservation of Mechanical Energy

The problem involves classical mechanics and energy conservation principles in Physics. To determine the angular speed of the hoop at the bottom of the ramp, we can utilize the conservation of mechanical energy. Initially, the hoop is at rest at the top of the ramp with specific dimensions and angles provided. The vertical drop height can be calculated using the length of the ramp and incline angle.

The conservation of energy equation is applied, equating the gravitational potential energy at the top to the kinetic energy at the bottom, including both translational and rotational components. Since the hoop rolls without slipping, the moment of inertia and the relationships for translational and rotational speed are considered in the calculations.

By substituting the values given for mass, radius, gravitational acceleration, ramp length, and incline angle into the equations, we can derive the final angular speed of the hoop at the bottom of the ramp.

Therefore, the angular speed of the hoop at the bottom of the ramp is determined to be 5.84 rad/s.