13. Rotational Inertia & Energy

A wheel with rotational inertia I = (1/2)MR² about its horizontal central axle is set spinning with initial angular speed ω₀. It is then lowered, and at the instant its edge touches the ground the speed of the axle (and cm) is zero. Initially the wheel slips when it touches the ground, but then begins to move forward and eventually rolls without slipping. How long does the wheel slip before it begins to roll without slipping? [Hint: Use $\sum \vec{F}=m\vec{a}$, $\sum {\tau }_{\mathrm{cm}}=I_{\mathrm{cm}}{\alpha }_{\mathrm{cm}}$, and recall that only when there is rolling without slipping is $v_{\mathrm{cm}}=\omega R$.]

(III) A wheel with rotational inertia I = 1/2 MR² about its horizontal central axle is set spinning with initial angular speed ω₀. It is then lowered, and at the instant its edge touches the ground the speed of the axle (and cm) is zero. Initially the wheel slips when it touches the ground, but then begins to move forward and eventually rolls without slipping. What is the wheel’s final translational speed, v_cm? [Hint: Use ${\sum }^{}\overrightarrow{F}=m\overrightarrow{a}$, $\sum {\tau }_{\mathrm{cm}}{\text{=I}}_{\mathrm{cm}}{\alpha }_{\mathrm{cm}}$, and recall that only when there is rolling without slipping is v_cm = ωR .]

A bicycle racer is going downhill at 11.0 m/s when, to his horror, one of his 2.25-kg wheels comes off as he is 75.0 m above the foot of the hill. We can model the wheel as a thin-walled cylinder 85.0 cm in diameter and ignore the small mass of the spokes. How much total kinetic energy does the wheel have when it reaches the bottom of the hill?

II) A uniform solid sphere of radius r₀ = 24.5 cm and mass m = 1.60 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long. Calculate its translational and rotational speeds when it reaches the bottom. Avoid putting in numbers until the end so you can answer.

An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls, without slipping, to the bottom. What is the sphere’s angular velocity at the bottom of the incline?