± Hoop on a Ramp

Learning Goal:

To apply the equations of general plane motion to describe the motion of a symmetric rigid body.

As shown, a circular hoop of mass m, radius r, and infinitesimal thickness rolls, without slipping, down a ramp that is inclined at an angle θ with the horizontal.

Part A

What is a, the acceleration of the hoop’s center?

Express your answer in terms of some or all of the variables m, θ, and g, the acceleration due to gravity.

a =

gsin(θ)2

Part B

What is the minimum coefficient of static friction, μmin, needed for the hoop to roll without slipping? Note that it is static and not kinetic friction that is relevant here, because the bottom point on the hoop is not moving relative to the ground (which is what it is meant by no slippage).

Express your answer in terms of all or some of the variables m, r, and θ.

μmin =

tan(θ)2

Part C

Imagine that the hoop is a tire. The coefficient of static friction between rubber and concrete is at least 0.9. What is θmax, the maximum angle that the ramp can have without the tire slipping?

Express your answer numerically in degrees to two significant figures.

θmax =

61

degrees