16. Angular Momentum

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.

(II) A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of 2.5 rev/s. If her initial moment of inertia was 4.9kg · m², what is her final moment of inertia? How does she physically accomplish this change?

(II) A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920. kg·m² . The platform rotates without friction with angular velocity 0.95 rad/s . The person walks radially to the edge of the platform.

(a) Calculate the angular velocity when the person reaches the edge.

(II) A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920. kg·m² . The platform rotates without friction with angular velocity 0.95 rad/s . The person walks radially to the edge of the platform.

(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.

(II) A uniform disk turns at 4.1 rev/s around a frictionless central axis. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 11–32. They then turn together around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?

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(II) A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω_0 about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform.

(a) Determine the angular velocity of the system as a function of time.

(II) A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω_0 about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform.

(b) What will be the angular velocity when the woman reaches the center?

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing 0.70 of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrown-off mass carries off either

(a) no angular momentum.

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing 0.70 of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrown-off mass carries off either

(b) its proportional share (0.70) of the initial angular momentum.

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

(c) If one rotation takes 1.2 s when the person’s arms are outstretched, what is the time for each rotation with arms at the sides? Ignore the moment of inertia of the lightweight platform.

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

One rotation takes 1.2 s when the person’s arms are outstretched. Ignore the moment of inertia of the lightweight platform.

(d) Determine the change in kinetic energy when the arms are lifted from the sides to the horizontal position.

(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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(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. 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.

(II) A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0.80 rad/s. Its total moment of inertia is 1630 kg·m². Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round.

(b) What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

CP A small block on a frictionless, horizontal surface has a mass of 0.0250 kg. It is attached to a massless cord passing through a hole in the surface (Fig. E10.40). The block is originally revolving at a distance of 0.300 m from the hole with an angular speed of 2.85 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.150 m. Model the block as a particle. (b) What is the new angular speed?

(III) Suppose a 65-kg person stands at the edge of a 9.8-m-diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1850 kg·m² . The turntable is at rest initially, but when the person begins running at a speed of 3.8 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.