Acceleration of a Rolling Hoop Down a Sloping Ramp

What is the acceleration of a hoop rolling down a sloping ramp at an angle?

What factors affect the acceleration of a hoop rolling down a sloping ramp without slipping?

Acceleration of a Rolling Hoop

The acceleration of a rolling hoop, of mass m and radius r, rolling down a ramp at an angle of 30 degrees without slipping, can be determined using principles of net force, torque, and rotational motion. The derived formula for the acceleration is a = (2/3)*g*sin(30), where g signifies acceleration due to gravity.

Explanation: The hoop rolling down a sloping ramp that makes an angle with the ground without slipping is a common scenario in physics, specifically dynamics. The movement involves both translational and rotational motion. To find the acceleration, we'll apply concepts from Newton's second law and the principles of rotational motion. The net force on the hoop as it rolls down the incline is the gravitational force component acting down the slope (equal to m*g*sin(angle)), and friction, which facilitates rolling without slipping. But in this instance, friction does not do any work as the point of contact is momentarily at rest.

Setting up Newton's second law for the hoop yields m*a = m*g*sin(30) (taking downhill as the positive direction). Now, for a hoop, the moment of inertia, I = m*r², and the net torque on the hoop due to gravity τ = I*alpha, where alpha is the angular acceleration. Considering the hoop rolls without slipping, the acceleration a = alpha*r.

Solving these equations, we get the acceleration of the hoop rolling down the incline as a = (2/3)*g*sin(30), where g is acceleration due to gravity.