Rolling Hoop Dynamics: Acceleration and Friction

A. What is the acceleration α of the center of the hoop?

B. What is the minimum coefficient of (static) friction μmin needed for the hoop to roll without slipping?

Answer:

The acceleration of a hoop rolling down a ramp is given by: a = g * sin(θ) / 2

The minimum coefficient of static friction needed for the hoop to roll without slipping is defined by: μmin = (1/2) * tan(θ)

Explanation:

Understanding the Behavior of a Rolling Hoop

The physical behavior of a rolling hoop can be described using the principles of Newton's Second Law and the conservation of angular momentum. These are used to derive the equations for acceleration and the required static friction coefficient.

Acceleration of the Center of the Hoop

The acceleration of the center of a hoop rolling down a ramp (α) can be calculated using the equation α = g * sin(θ) / (1 + I/mr²), where g is the acceleration due to gravity, θ is the incline angle, and I is the moment of inertia of the hoop. For a hoop, I = m*r². So, the acceleration (α) becomes α = g * sin(θ) / 2.

Minimum Coefficient of Static Friction

The minimum coefficient of static friction (μmin) ensures that the hoop rolls without slipping. It can be calculated using the equation: μmin = (1/2) * tan(θ). This equation demonstrates that the friction required depends on the angle of inclination θ of the ramp.