16. Angular Momentum

Conservation of Angular Momentum

Open Question
During most of its lifetime, a star maintains an equilibrium size in which the inward force of gravity on each atom is balanced by an outward pressure force due to the heat of the nuclear reactions in the core. But after all the hydrogen 'fuel' is consumed by nuclear fusion, the pressure force drops and the star undergoes a gravitational collapse until it becomes a neutron star. In a neutron star, the electrons and protons of the atoms are squeezed together by gravity until they fuse into neutrons. Neutron stars spin very rapidly and emit intense pulses of radio and light waves, one pulse per rotation. These 'pulsing stars' were discovered in the 1960s and are called pulsars.a. A star with the mass (M = 2.0 X 10^30 kg) and size (R = 7.0 x 10^8 m) of our sun rotates once every 30 days. After undergoing gravitational collapse, the star forms a pulsar that is observed by astronomers to emit radio pulses every 0.10 s. By treating the neutron star as a solid sphere, deduce its radius.
Open Question
(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.
Open Question
(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.
Open Question
(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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Open Question
(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.
Open Question
(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?
Open Question
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.
Open Question
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.
Open Question
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.
Open Question
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.
Open Question
(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?
Open Question
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?
Open Question
(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>
Open Question
(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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Open Question
(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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