(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.
(a) In what direction does friction act on the slipping wheel? [Hint: Use ∑ F (→ above F) = m a (→ above a), ∑ ₜ꜀ₘα꜀ₘ , and recall that only when there is rolling without slipping is v꜀ₘ = ωR .]

(II) A thin, hollow 0.545-kg section of pipe of radius 12.0 cm starts rolling (from rest) down a 17.5° incline 5.60 m long.

(a) If the pipe rolls without slipping, what will be its speed at the base of the incline?

(II) A thin, hollow 0.545-kg section of pipe of radius 12.0 cm starts rolling (from rest) down a 17.5° incline 5.60 m long.

(b) What will be its total kinetic energy at the base of the incline?

(II) A ball of radius r₀ rolls on the inside of a track of radius R₀ (see Fig. 10–68). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping the whole way?

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(a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.013 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest. (b) What fraction of its kinetic energy is rotational?

(II) A thin, hollow 0.545-kg section of pipe of radius 12.0 cm starts rolling (from rest) down a 17.5° incline 5.60 m long.

(c) What minimum value must the coefficient of static friction have if the pipe is not to slip?

A 5.0-m-long ladder is leaning against the side of a building making a 35° angle with the building. When a person is about 1/3 of the way up, the ladder slips and falls to the ground in 3.0 s. What is the average angular acceleration of the ladder as it falls?

(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.

(b) How long does the wheel slip before it begins to roll without slipping? [Hint: Use ∑ F (→ above F) = m a (→ above a), ∑ ₜ꜀ₘα꜀ₘ , and recall that only when there is rolling without slipping is v꜀ₘ = ωR .]

(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.

(c) What is the wheel’s final translational speed, v꜀ₘ? [Hint: Use ∑ F (→ above F) = m a (→ above a), ∑ ₜ꜀ₘα꜀ₘ , and recall that only when there is rolling without slipping is v꜀ₘ = ωR .]