(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v₀ and ω₀ a “reverse” spin of angular speed (see Fig. 11–41). A kinetic friction force acts on the ball as it initially skids across the table.
(d) If ω₀ is 10% larger than ,w_C i.e.,ω₀ = 1.10w_C, determine the ball’s cm velocity v_CM when it starts to roll without slipping. [Hint: The ball possesses two types of angular momentum, the first due to the linear speed v_CM of its cm relative to point O, the second due to the spin at angular velocity ω about its own cm. The ball’s total L about O is the sum of these two angular momenta.]
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(II) Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia.

(c) If they now pull on each other’s hands, reducing their radius to half its original value, what is their common angular speed after reducing their radius?

A 70.0-kg person stands on a tiny rotating platform with arms outstretched.

(e) From your answer to part (d), would you expect it to be harder or easier to lift your arms when rotating or when at rest?

(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v₀ and ω₀ a “reverse” spin of angular speed (see Fig. 11–41). A kinetic friction force acts on the ball as it initially skids across the table.

(b) Using conservation of angular momentum, find the critical angular speed ω_C such that, if ω₀=ω_C, kinetic friction will bring the ball to a complete (as opposed to momentary) stop. <IMAGE>

(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v₀ and ω₀ a “reverse” spin of angular speed (see Fig. 11–41). A kinetic friction force acts on the ball as it initially skids across the table.

(c) If ω₀ is 10% smaller than ω_C , i.e., ω₀ = 0.90ω_C , determine the ball’s cm velocity v_CM when it starts to roll without slipping.

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(II) Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia. They now pull on each other’s hands, reducing their radius to half its original value

(d) Calculate the change in kinetic energy for this process.

Suppose a diver spins at 8 rad/s while falling with a moment of inertia about an axis through himself of 3 kg•m². What moment of inertia would the diver need to have to spin at 4 rad/s? 

BONUS:How could he accomplish this?

Two astronauts, both 80 kg, are connected in space by a light cable. When they are 10 m apart, they spin about their center of mass with 6 rad/s. Calculate the new angular speed they'll have if they pull on the rope to reduce their distance to 5 m. You may treat them as point masses, and assume they continue to spin around their center of mass.