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.)

(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 .]

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.

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