13. Rotational Inertia & Energy

A solid sphere of mass M = 10 kg and radius R = 2 is rolling without slipping with speed V = 5 m/s on a flat surface when it reaches the bottom of an inclined plane that makes an angle of Θ = 37° with the horizontal. The plane has just enough friction to cause the sphere to roll without slipping while going up. What maximum height will the sphere attain? (Use g = 10 m/s².)

You may remember that the lowest speed that an object may have at the top of a loop-the-loop of radius R, so that it completes the loop without falling, is √gR . Determine the lowest speed that a solid sphere must have at the bottom of a loop-the-loop, so that it reaches the top with enough speed to complete the loop. Assume the sphere rolls without slipping.

(I) Calculate the translational speed of a uniform solid cylinder when it reaches the bottom of an incline 6.50 m high. Assume it starts from rest and rolls without slipping.

A marble of mass m and radius r rolls along the looped rough track of Fig. 10–77. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? (a) Assume r << R ; (b) do not make this assumption. Ignore frictional losses.

<IMAGE>

(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?

<IMAGE>

(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?

(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.

(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 .]