16. Angular Momentum

Multiple Choice

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.

The rotor (flywheel) of a toy gyroscope has mass 0.140 kg. Its moment of inertia about its axis is 1.20 * 10^-4 kg•m^2. The mass of the frame is 0.0250 kg. The gyroscope is supported on a single pivot (Fig. E10.51) with its center of mass a horizontal distance of 4.00 cm from the pivot. The gyroscope is precessing in a horizontal plane at the rate of one revolution in 2.20 s.

(a) Find the upward force exerted by the pivot.

544

Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly 10^14 times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star's initial radius was 7.0 * 10^5 km (comparable to our sun); its final radius is 16 km. If the original star rotated once in 30 days, find the angular speed of the neutron star.

A thin uniform rod has a length of 0.500 m and is rotating in a circle on a frictionless table. The axis of rotation is perpendicular to the length of the rod at one end and is stationary. The rod has an angular velocity of 0.400 rad/s and a moment of inertia about the axis of 3.00 * 10-3 kg/m^2. A bug initially standing on the rod at the axis of rotation decides to crawl out to the other end of the rod. When the bug has reached the end of the rod and sits there, its tangential speed is 0.160 m/s. The bug can be treated as a point mass. What is the mass of (a) the rod;

A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. (b) Compute the kinetic energy of the system before and after the parachutist lands. Why are these kinetic energies not equal?

971

A Gyroscope on the Moon. A certain gyroscope precesses at a rate of 0.50 rad/s when used on earth. If it were taken to a lunar base, where the acceleration due to gravity is 0.165g, what would be its precession rate?

339

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. (d) How much work was done in pulling the cord?

573

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. (c) Find the change in kinetic energy of the block.

The Spinning Figure Skater. The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center (Fig. E10.43). When the skater's hands and arms are brought in and wrapped around his body to execute the spin, the hands and arms can be considered a thinwalled, hollow cylinder. His hands and arms have a combined mass of 8.0 kg. When outstretched, they span 1.8 m; when wrapped, they form a cylinder of radius 25 cm. The moment of inertia about the rotation axis of the remainder of his body is constant and equal to 0.40 kg•m2 . If his original angular speed is 0.40 rev/s, what is his final angular speed?

2789

A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. (a) Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.)

540

(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?

40

A 10 g bullet traveling at 400 m/s strikes a 10 kg, 1.0-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door just after impact?

179

A satellite follows the elliptical orbit shown in FIGURE P12.77. The only force on the satellite is the gravitational attraction of the planet. The satellite's speed at point 1 is 8000 m/s. b. What is the satellite's speed at point 2?

205

A merry-go-round is a common piece of playground equipment. A 3.0-m-diameter merry-go-round with a mass of 250 kg is spinning at 20 rpm. John runs tangent to the merry-go-round at 5.0 m/s, in the same direction that it is turning, and jumps onto the outer edge. John's mass is 30 kg. What is the merry-go-round's angular velocity, in rpm, after John jumps on?

571

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.

134

(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.

50

(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.

31

(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?

<IMAGE>

41

(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.

19

(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?

41

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.

39

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.

34

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.

45

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.

48

(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?

41

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?

38

(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>

31

(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.

<IMAGE>

38

(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.]

<IMAGE>

42

(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.

50

"(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.

(a) What is the angular velocity of the merry-go-round now?"

16

(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)?

16

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. (a) Is the angular momentum of the block conserved? Why or why not?

15