5. A hollow cylinder of mass m, radius Rc, and moment of inertia I = mRc2 is pushed against a spring (with spring constant k) compressing it by a distance d. It is then released and rolls without slipping on a track, and through a vertical loop of radius RL. Assume RC << RL (a) When the cylinder reaches the top of the vertical loop, what is the minimum (linear) speed it must have to avoid falling off? Draw a free-body diagram to support your answer.
(b) What is the minimum compression of the spring necessary to prevent the cylinder falling off?
(c) Perform two “Cross-checks” on you solutions to check the validity of your solution, and/or examine the behavior of the system.

Explanation:

(a) Draw a free body diagram of the cylinder at the top of the loop.  At the minimum speed, the normal force is 0, so the only force is weight pulling down.

Sum of forces in the centripetal direction:

∑F = ma

mg = mv²/RL

v = √(g RL)

(b) Energy is conserved.

EE = KE + RE + PE

½ kd² = ½ mv² + ½ Iω² + mgh

kd² = mv² + Iω² + 2mgh

kd² = mv² + (m RC²) ω² + 2mg (2 RL)

kd² = mv² + m RC²ω² + 4mg RL

kd² = mv² + mv² + 4mg RL

kd² = 2mv² + 4mg RL

kd² = 2m (v² + 2g RL)

d² = 2m (v² + 2g RL) / k

d = √[2m (v² + 2g RL) / k]

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