Using the vector product, prove the law of sines for the

True or false: (a) If two vectors are exactly opposite in direction, their vector product must be zero. (b) The magnitude of the vector product of two vectors is at a minimum when the vectors are perpendicular. (c) Knowing the magnitude of the vector product of two nonzero vectors and the vectors individual magnitudes uniquely determines the angle between them.

Consider two nonzero vectors \(\vec{A}\) and \(\vec{B}\). Their vector product has the greatest magnitude if \(\vec{A}\) and \(\vec{B}\) are (a) parallel, (b) perpendicular, (c) antiparallel, (d) at an angle of \(45^{\circ}\) to each other.

What is the angle between a force vector and a torque vector generated by

A point particle of mass m is moving with a constant speed along a straight line that passes through point P. What can you say about the angular momentum of the particle relative to point P? (a) Its magnitude is mv. (b) Its magnitude is zero. (c) Its magnitude changes sign as the particle passes through point P. (d) Its magnitude increases as the particle approaches point P.

A particle travels in a circular path, and point P is at the center of the circle. (a) If the particles linear momentum is doubled without changing the radius of the circle, how is the magnitude of its angular momentum about P affected? (b) If the radius of the circle is doubled but the speed of the particle is unchanged, how is the magnitude of its angular momentum about P affected?

A particle moves along a straight line at constant speed. How does the particles angular momentum about any fixed point vary with time?

True or false: If the net torque on a rotating object is zero, the angular velocity of the object cannot change. If your answer is false, give an example of such a situation.

You are standing on the edge of a turntable with frictionless bearings that is initially rotating when you catch a ball that is moving in the same direction but faster than you are moving and on a line tangent to the edge of the turntable. Assume you do not move relative to the turntable. (a) Does the angular speed of the turntable increase, decrease, or remain the same during the catch? (b) Does the magnitude of your angular momentum (about the rotation axis of the table) increase, decrease, or remain the same during the catch? (c) How does the balls angular momentum (about the rotation axis of the table) change after the catch? (d) How does the total angular momentum of the system, youtableball (above the rotation axis of the turntable), change after the catch?

If the angular momentum of a system about a fixed point P is constant, which one of the following statements must be true? (a) No torque about P acts on any part of the system. (b) A constant torque about P acts on each part of the system. (c) Zero net torque about P acts on each part of the system. (d) A constant external torque about P acts on the system. (e) Zero net external torque about P acts on the system.

A block sliding on a frictionless table is attached to a string that passes through a narrow hole through the tabletop. Initially, the block is sliding with speed in a circle of radius A student under the table pulls slowly on the string. What happens as the block spirals inward? Give supporting arguments for your choice. (The term angular momentum refers to the angular momentum about a vertical axis through the hole.) (a) Its energy and angular momentum are conserved. (b) Its angular momentum is conserved and its energy increases. (c) Its angular momentum is conserved and its energy decreases. (d) Its energy is conserved and its angular momentum increases. (e) Its energy is conserved and its angular momentum decreases.

One way to tell if an egg is hardboiled or uncooked without breaking the egg is to lay the egg flat on a hard surface and try to spin it. A hardboiled egg will spin easily, an uncooked egg will not. However, once spinning, the uncooked egg may do something unusual: If you stop it with your finger, it may start spinning again. Explain the difference in the behavior of the two types of eggs.

Explain why a helicopter with just one main rotor has a second smaller rotor mounted on a horizontal axis at the rear, as in Figure 10-40. Describe the resultant motion of the helicopter if this rear rotor fails during flight.

The spin angular-momentum vector for a spinning wheel is parallel with its axle and is pointed east. To cause this vector to rotate toward the south, in which direction must a force be exerted on the east end of the axle? (a) up, (b) down, (c) north, (d) south, (e) east.

CONTEXT-RICH You are walking toward the north, and in your left hand you are carrying a suitcase that contains a massive spinning wheel mounted on an axle attached to the front and back of the case. The angular velocity of the gyroscope points north. You now begin to turn to walk toward the east. As a result, the front end of the suitcase will (a) resist your attempt to turn it and will try to maintain its original orientation, (b) resist your attempt to turn and will pull to the west, (c) rise upward, (d) dip downward, (e) show no effect whatsoever.

ENGINEERING APPLICATION The angular momentum of the propeller of a small single-engine airplane points forward. The propeller rotates clockwise if viewed from behind. (a) Just after liftoff, as the nose of the plane tilts upward, the airplane tends to veer to one side. To which side does it tend to veer and why? (b) If the plane is flying horizontally and suddenly turns to the right, does the nose of the plane tend to veer upward or downward? Why?

CONTEXT-RICH, ENGINEERING APPLICATION You have designed a car that is powered by the energy stored in a single flywheel with a spin angular momentum In the morning, you plug the car into an electrical outlet and a motor spins the flywheel up to speed, adding a huge amount of rotational kinetic energy to itenergy that will be changed into translational kinetic energy of the car during the day. Having taken a physics course involving angular momentum and torques, you realize that problems would arise during various maneuvers of the car. Discuss some of these problems. For example, suppose the flywheel is mounted so that points vertically upward when the car is on a horizontal road. What would happen as the car travels over a hilltop? Through a valley? Suppose the flywheel is mounted so that points forward or to one side when the car is on a horizontal road. Then what would happen if the car attempts to turn to the left or right? In each case that you examine, consider the direction of the torque exerted on the car by the road.

You are sitting on a spinning piano stool with your arms folded. (a) When you extend your arms out to your sides, what happens to your kinetic energy? What is the cause of this change? (b) Explain what happens to your moment of inertia, angular speed, and angular momentum as you extend your arms.

A uniform rod of mass M and length L rests on a horizontal frictionless table. A blob of putty of mass moves along a line perpendicular to the rod, strikes the rod near its end, and sticks to the rod. Describe qualitatively the subsequent motion of the rod and putty.

An ice-skater starts her pirouette with arms outstretched, rotating at Estimate her rotational speed (in revolutions per second) when she brings her arms flat against her body.

Estimate the ratio of angular velocities for the rotation of a diver between the full tuck position and the full layout position.

The days on Mars and Earth are of nearly identical length. Earths mass is 9.35 times Marss mass, Earths radius is 1.88 times Marss radius, and Mars is on average 1.52 times farther away from the Sun than Earth is. The Martian year is 1.88 times longer than Earths year. Assume that they are both uniform spheres and that their orbits about the Sun are circles. Estimate the ratio (Earth to Mars) of (a) their spin angular momenta, (b) their spin kinetic energies, (c) their orbital angular momenta, and (d) their orbital kinetic energies.

The polar ice caps contain about of ice. This mass contributes negligibly to the moment of inertia of Earth because it is located at the poles, close to the axis of rotation. Estimate the change in the length of the day that would be expected if the polar ice caps were to melt and the water were distributed uniformly over the surface of Earth.

A 2.0-g particle moves at a constant speed of around a circle of radius 4.0 mm. (a) Find the magnitude of the angular momentum of the particle. (b) If where is an integer, find the value of and the approximate value of (c) By how much does change if the particles speed increases by one-millionth of a percent, and nothing else changes? Use your result to explain why the quantization of angular momentum is not noticed in macroscopic physics.

Astrophysicists in the 1960s tried to explain the existence and structure of pulsarsextremely regular astronomical sources of radio pulses whose periods ranged from seconds to milliseconds. At one point, these radio sources were given the acronym LGM (Little Green Men), a reference to the idea that they might be signals from extraterrestrial civilizations. The explanation given today is no less interesting. Consider the following. Our Sun, which is a fairly typical star, has a mass of and a radius of Although it does not rotate uniformly, because it is not a solid body, its average rate of rotation is about Stars larger than the Sun can expire in spectacular explosions called supernovae, leaving behind a collapsed remnant of the star called a neutron star. These neutron-stars have masses comparable to the original masses of the stars but radii of only a few kilometers! The high rotation rate are due to the conservation of angular momentum during the collapses. These stars emit beams of radio waves. Because of the rapid angular speed of the stars, the beam sweeps past Earth at regular, very short, intervals. To produce the observed radio-wave pulses, the star has to rotate at rates that range from about to (a) Using data from the textbook, estimate the rotation rate of the Sun if it were to collapse into a neutron star of radius The Sun is not a uniform sphere of gas, and its moment of inertia is given by Assume that the neutron star is spherical and has a uniform mass distribution. (b) Is the rotational kinetic energy of the Sun greater or smaller after the collapse? By what factor does it change, and where does the energy go to or come from?

The moment of inertia of Earth about its spin axis is approximately (a) Because Earth is nearly spherical, assume that the moment of inertia can be written as where C is a dimensionless constant, is the mass of Earth, and is its radius. Determine C. (b) If Earths mass were distributed uniformly, C would equal From the value of C calculated in Part (a), is Earths density greater near its center or near its surface? Explain your reasoning.

Estimate Timothy Goebels initial takeoff speed, rotational velocity, and angular momentum when he performs a quadruple lutz (Figure 10-41). Make any assumptions you think reasonable, but justify them. Goebels mass is about and the height of the jump is about 0.60m, Note that his angular speed will change quite a bit during the jump, as he begins with arms outstretched and then pulls them in. Your answer should be accurate to within a factor of 2, if you are careful.

A force of magnitude F is applied horizontally in the direction to the rim of a disk of radius as shown in Figure 10-42. Write and in terms of the unit vectors and and compute the torque produced by this force about the origin at the center of the disk. SSM

Compute the torque about the origin of the gravitational force acting on a particle of mass m located at and show that this torque is independent of the y coordinate.

Find for the following choices: (a) and (b) and and (c) and

For each case in Problem 29, compute Compare it to to estimate which of the pairs of vectors are closest to being perpendicular. Verify your answers by calculating the angle using the dot product.

A particle moves in a circle that is centered at the origin. The particle has position and angular velocity (a) Show that its velocity is given by (b) Show that its centripetal acceleration is given by

You are given three vectors and their components in the form: and Show that the following equalities hold:

If and find

If and determine

Given three noncoplanar vectors, and show that is the volume of the parallelepiped formed by the three vectors.

Using the vector product, prove the law of sines for the triangle shown in Figure 10-43. That is, if A, B, and C are the lengths of each side of the triangle, show that A>sin a _ B>sin b _ C>sin c.

A2.0-kg particle moves directly eastward at a constant speed of along an eastwest line. (a) What is its angular momentum (including direction) about a point that lies north of the line? (b) What is its angular momentum (including direction) about a point that lies south of the line? (c) What is its angular momentum (including direction) about a point that lies directly east of the particle?

You observe a 2.0-kg particle moving at a constant speed of in a clockwise direction around a circle of radius (a) What is its angular momentum (including direction) about the center of the circle? (b) What is its moment of inertia about an axis through the center of the circle and perpendicular to the plane of the motion? (c) What is the angular velocity of the particle?

(a) A particle moving at constant velocity has zero angular momentum about a particular point. Use the definition of angular momentum to show that under this condition the particle is moving either directly toward or directly away from the point. (b) You are a right-handed batter and let a waist-high fastball go past you without swinging. What is the direction of the balls angular momentum relative to your navel? (Assume that the ball travels in a straight, horizontal line as it passes you.)

A particle that has a mass m is traveling with a constant velocity along a straight line that is a distance b from the origin O (Figure 10-44). Let be the area swept out by the position vector from O to the particle during a time interval dt. Show that is constant and is equal to where L is the magnitude of the angular momentum of the particle about the origin.

A 15-g coin that has a diameter equal to is spinning at about a fixed vertical axis. The coin is spinning on edge with its center directly above the point of contact with the tabletop. As you look down on the tabletop, the coin spins clockwise. (a) What is the angular momentum (including direction) of the coin about its center of mass? (To find the moment of inertia about the axis, see Table 9-1.) Model the coin as a cylinder of length L and take the limit as L approaches zero. (b) What is the coins angular momentum (including direction) about a point on the tabletop from the axis? (c) Now the coins center of mass travels at in a straight line east across the tabletop, while spinning the same way as in Part (a). What is the angular momentum (including direction) of the coin about a point on the line of motion of the center of mass? (d) When it is both spinning and sliding, what is the angular momentum of the coin (including direction) about a point north of the line of motion of the center of mass?

CONCEPTUAL (a) Two stars of masses and are located at and relative to some origin O, as shown in Figure 10-45. They exert equal and opposite attractive gravitational forces on each other. For this two-star system, calculate the net torque exerted by these internal forces about the origin O and show that it is zero only if both forces lie along the line joining the particles. (b) The fact that the Newtons third-law pair of forces are not only equal and oppositely directed but also lie along the line connecting the two objects is sometimes called the strong form of Newtons third law. Why is it important to add that last phrase? Hint: Consider what would happen to these two objects if the forces were offset from each other.

A 1.8-kg particle moves in a circle of radius As you look down on the plane of its orbit, the particle is initially moving clockwise. If we call the clockwise direction positive, the particles angular momentum relative to the center of the circle varies with time according to (a) Find the magnitude and direction of the torque acting on the particle. (b) Find the angular velocity of the particle as a function of time.

CONTEXT-RICH, ENGINEERING APPLICATION You are designing a lathe motor, part of which consists of a uniform cylinder whose mass is and whose radius is that is mounted so that it turns without friction on its axis, which is fixed. The cylinder is driven by a belt that wraps around its perimeter and exerts a constant torque. At the cylinders angular velocity is zero. At its angular speed is (a) What is the magnitude of the cylinders angular momentum at (b) At what rate is the angular momentum increasing? (c) What is the magnitude of the torque acting on the cylinder? (d) What is the magnitude of the frictional force acting on the rim of the cylinder?

In Figure 10-46, the incline is frictionless and the string passes through the center of mass of each block. The pulley has a moment of inertia I and radius R. (a) Find the net torque acting on the system (the two masses, string, and pulley) about the center of the pulley. (b) Write an expression for the total angular momentum of the system about the center of the pulley. Assume the masses are moving with a speed . (c) Find the acceleration of the masses by using your results for Parts (a) and (b) and by setting the net torque equal to the rate of change of the systems angular momentum. SSM

CONTEXT-RICH, ENGINEERING APPLICATION Figure 10-47 shows the rear view of a space capsule that was left rotating rapidly about its axis at after a collision with another capsule. You are the flight controller and have just moments to tell the crew how to stop this rotation before they become ill from the rotation and the situation becomes dangerous. You know that they have access to two small jets mounted tangentially at a distance from the axis, as indicated in the figure. These jets can each eject of gas with a nozzle speed of Determine the length of time these jets must run to stop the rotation. In flight, the moment of inertia of the ship about its axis (assumed constant) is known to be 4000 kg # m2.

A projectile (mass M) is launched at an angle with an initial speed Considering the torque and angular momentum about the launch point, explicitly show that Ignore the effects of air resistance. (The equations for projectile motion are found in Chapter 3.)

A planet moves in an elliptical orbit about the Sun, with the Sun at one focus of the ellipse, as in Figure 10-48. (a) What is the torque about the center of the Sun due to the gravitational force of attraction of the Sun on the planet? (b) At position A, the planet has an orbital radius and is moving with a speed perpendicular to the line from the Sun to the planet. At position B, the planet has an orbital radius and is moving with speed again perpendicular to the line from the Sun to the planet. What is the ratio of to in terms of and

You stand on a frictionless platform that is rotating at an angular speed of Your arms are outstretched, and you hold a heavy weight in each hand. The moment of inertia of you, the extended weights, and the platform is When you pull the weights in toward your body, the moment of inertia decreases to (a) What is the resulting angular speed of the platform? (b) What is the change in kinetic energy of the system? (c) Where did this increase in energy come from?

A small blob of putty of mass m falls from the ceiling and lands on the outer rim of a turntable of radius R and moment of inertia that is rotating freely with angular speed about its vertical fixed-symmetry axis. (a) What is the postcollision angular speed of the turntableputty system? (b) After several turns, the blob flies off the edge of the turntable. What is the angular speed of the turntable after the blobs departure? 51 Alazy Susan consists of a heavy plastic cylinder mounted

a frictionless bearing resting on a vertical shaft through its center. The cylinder has a radius and mass A cockroach (mass ) is on the lazy Susan, at a distance of from the center. Both the cockroach and the lazy Susan are initially at rest. The cockroach then walks along a circular path concentric with the axis of the lazy Susan at a constant distance of from the axis of the shaft. If the speed of the cockroach with respect to the lazy Susan is what is the speed of the cockroach with respect to the room? SSM

Two disks of identical mass but different radii (r and 2r) are spinning on frictionless bearings at the same angular speed but in opposite directions (Figure 10-49). The two disks are brought slowly together. The resulting frictional force between the surfaces eventually brings them to a common angular velocity. (a) What is the magnitude of that final angular velocity in terms of (b) What is the change in rotational kinetic energy of the system? Explain.

A block of mass m sliding on a frictionless table is attached to a string that passes through a narrow hole through the center of the table. The block is sliding with speed in a circle of radius r Find (a) the angular momentum of the block, (b) the 0 . kinetic energy of the block, and (c) the tension in the string. (d) A student under the table now slowly pulls the string downward. How much work is required to reduce the radius of the circle from to

A 0.20-kg point mass moving on a frictionless horizontal surface is attached to a rubber band whose other end is fixed at point P. The rubber band exerts a force whose magnitude is where x is the length of the rubber band and b is an unknown constant. The rubber band force points inward toward P. The mass moves along the dotted line in Figure 10-50. When it passes point A, its velocity is directed as shown. The distance AP is and BP is (a) Find the speed of the mass at points B and C. (b) Find b.

The z component of the spin of an electron is but the magnitude of the spin vector is What is the angle between the electrons spin angular-momentum vector and the positive z axis?

Show that the energy difference between one rotational state of a molecule and the next higher state is proportional to

CONTEXT-RICH, BIOLOGICAL APPLICATION You work in a biochemical research lab, where you are investigating the rotational energy levels of the HBr molecule. After consulting the periodic chart, you know that the mass of the bromine atom is 80 times that of the hydrogen atom. Consequently, in calculating the rotational motion of the molecule, you assume, to a good approximation, that the Br nucleus remains stationary as the H atom revolves around it. You also know that the separation between the hydrogen atom and bromine nucleus is Calculate (a) the moment of inertia of the HBr molecule about the bromine nucleus, and (b) the rotational energies for the bromine nucleuss ground state (lowest energy) and the next two states of higher energy (called the first and second excited states) described by and

The equilibrium separation between the nuclei of the nitrogen molecule is and the mass of each nitrogen nucleus is where For rotational energies, the total energy is due to rotational kinetic energy. (a) Approximate the nitrogen molecule as a rigid dumbbell of two equal point masses and calculate the moment of inertia about its center of mass. (b) Find the energy of the lowest three energy levels using (c) Molecules emit a particle (or quantum) of light called a photon when they make a transition from a higher energy state to a lower one. Determine the energy of a photon emitted when a nitrogen molecule drops from the to the state. Visible-light photons each have between 2 and 3 eV of energy. Are these photons in the visible region?

CONCEPTUAL Consider a transition from a lower energy state to a higher onethat is, the absorption of a quantum of energy, resulting in an increase in the rotational energy of an molecule (see Problem 58). Suppose such a molecule, initially in its ground rotational state, was exposed to photons, each with an energy equal to three times the energy of its first excited state. (a) Would the molecule be able to absorb this photon energy? Explain why or why not, and if it can, determine the energy level to which it goes. (b) To make a transition from its ground state to its second excited state requires how many times the energy of the first excited state?

A 16.0-kg, 2.40-m-long rod is supported at its midpoint on a knife edge. A 3.20-kg ball of clay drops from rest from a height of and makes a perfectly inelastic collision with the rod from the point of support (Figure 10-51). Find the angular momentum of the rod and clay system about the point of support immediately after the inelastic collision.

Figure 10-52 shows a thin, uniform bar of length L and mass M and a small blob of putty of mass m. The system is supported by a frictionless horizontal surface. The putty moves to the right with velocity strikes the bar at a distance d from the center of the bar, and sticks to the bar at the point of contact. Obtain expressions for the velocity of the systems center of mass and for the angular speed following the collision. SSM

Figure 10-52 shows a thin, uniform bar whose length is L and mass is Mand a compact hard sphere whose mass is m. The system is supported by a frictionless horizontal surface. The sphere moves to the right with velocity and strikes the bar at a distance from the center of the bar. The collision is elastic, and following the collision the sphere is at rest. Find the value of the ratio m>M.

Figure 10-53 shows a uniform rod whose length is L and mass is Mpivoted at the top. The rod, which is initially at rest, is struck by a particle whose mass is m at a point below the pivot. Assume that the particle sticks to the rod. What must be the speed of the particle so that following the collision the maximum angle between the rod and the vertical is 90?

If, for the system of Problem 63, and the maximum angle between the rod and the vertical following the collision is find the speed of the particle before impact.

A uniform rod is resting on a frictionless table when it is suddenly struck at one end by a sharp horizontal blow in a direction perpendicular to the rod. The mass of the rod is M and the magnitude of the impulse applied by the blow is J. Immediately after the rod is struck, (a) what is the velocity of the center of mass of the rod, (b) what is the velocity of the end that is struck, (c) and what is the velocity of the other end of the rod? (d) Is there a point on the rod that remains motionless?

A projectile of mass is traveling at a constant velocity toward a stationary disk of mass M and radius R that is free to rotate its axis (Figure 10-54). Before impact, the projectile is traveling along a line displaced a distance b below the axis. The projectile strikes the disk and sticks to point B. Model the projectile as a point mass. (a) Before impact, what is the total angular momentum of the diskprojectile system about the axis? Answer the following questions in terms of the symbols given at the start of this problem. (b) What is the angular speed v of the diskprojectile system just after the impact? (c) What is the kinetic energy of the diskprojectile system after impact? (d) How much mechanical energy is lost in this collision?

A uniform rod of length and mass M equal to is attached to a hinge of negligible mass at one end and is free to rotate in the vertical plane (Figure 10-55). The rod is released from rest in the position shown. A particle of mass is suspended from a thin string of length from the hinge. The particle sticks to the rod on contact. What should the ratio be so that umax after the collision? SSM

A uniform rod that has a length equal to and a mass equal to is attached to a hinge of negligible mass at one end and is free to rotate in the vertical plane (Figure 10-55). The rod is released from rest in the position shown. A particle whose mass is m is suspended from a thin string whose length is equal to from the hinge. The particle sticks to the rod on contact, and after the collision the rod continues to rotate until (a) Find m. (b) How much energy is dissipated during the collision? SSM

A bicycle wheel that has a radius equal to is mounted at the middle of a 50-cm-long axle. The tire and rim weigh The wheel is spun at and the axle is then placed in a horizontal position with one end resting on a pivot. (a) What is the angular momentum due to the spinning of the wheel? (Treat the wheel as a hoop.) (b) What is the angular velocity of precession? (c) How long does it take for the axle to swing through around the pivot? (d) What is the angular momentum associated with the motion of the center of mass, that is, due to the precession? In what direction is this angular momentum?

A uniform disk, whose mass is and radius is is mounted at the center of a 10.0-cm-long axle and spun at The axle is then placed in a horizontal position with one end resting on a pivot. The other end is given an initial horizontal velocity such that the precession is smooth with no nutation. (a) What is the angular velocity of precession? (b) What is the speed of the center of mass during the precession? (c) What is the acceleration (magnitude and direction) of the center of mass? (d) What are the vertical and horizontal components of the force exerted by the pivot on the axle?

A particle whose mass is moves in the xy plane with velocity along the line (a) Find the angular momentum about the origin when the particle is at ( ). (b) A force is applied to the particle. Find the torque about the origin due to this force as the particle passes through the point (12 m, 5.3 m).

The position vector of a particle whose mass is is given by where is in meters and t is in seconds. Determine the angular momentum and net torque about the origin acting on the particle.

Two ice-skaters, whose masses are and hold hands and rotate about a vertical axis that passes between them, making one revolution in Their centers of mass are separated by and their center of mass is stationary. Model each skater as a point particle and find (a) the angular momentum of the system about their center of mass and (b) the total kinetic energy of the system.

A 2.0-kg ball attached to a string whose length is moves counterclockwise (as viewed from above) in a horizontal circle (Figure 10-56). The string makes an angle with the vertical. (a) Determine both the horizontal and vertical components of the angular momentum of the ball about the point of support. (b) Find the magnitude of and verify that it equals the magnitude of the torque exerted by gravity about the point of support.

A compact object whose mass is m resting on a horizontal, frictionless surface is attached to a string that wraps around a vertical cylindrical post attached to the surface. Thus, when the object is set into motion, it follows a path that spirals inward. (a) Is the angular momentum of the object about the axis of the post conserved? Explain your answer. (b) Is the energy of the object condL served? Explain your answer. (c) If the speed of the object is when the unwrapped length of the string is r, what is its speed when the unwrapped length has shortened to

CONTEXT-RICH, ENGINEERING APPLICATION Figure 10-57 shows a hollow cylindrical tube that has a mass M, a length L, and a moment of inertia Inside the cylinder are two disks, each of mass m and radius r, separated by a distance , and tied to a central post by a thin string. The system can rotate about a vertical axis through the center of the cylinder. You are designing this cylinder disk apparatus to shut down the rotations by triggering an electronic shutoff signal (sent to the rotating motor) when the strings break as the disks hit the ends of the cylinder. During development, you notice that with the system rotating at some critical angular speed the string suddenly breaks. When the disks reach the ends of the cylinder, they stick. Obtain expressions for the final angular speed and the initial and final kinetic energies of the system. Assume that the inside walls of the cylinder are frictionless.

CONTEXT-RICH, ENGINEERING APPLICATION Repeat Problem 76, but this time friction between the disks and the walls of the cylinder is not negligible. However, the coefficient of friction is not great enough to prevent the disks from reaching the ends of the cylinder. Can the final kinetic energy of the system be determined without knowing the coefficient of kinetic friction?

CONTEXT-RICH, ENGINEERING APPLICATION Suppose that in Figure 10-57 and The string breaks when the systems angular speed approaches a critical angular speed at which time the tension in the string approaches 108 The masses then move radially outward until they undergo perfectly inelastic collisions with the ends of the cylinder. Determine the critical angular speed and the angular speed of the system after the inelastic collisions. Find the total kinetic energy of the system at the critical angular speed and again after the inelastic collisions. Assume that the inside walls of the cylinder are frictionless.

Keplers second law states: The line from the center of the Sun to the center of a planet sweeps out equal areas in equal times. Show that this law follows directly from the law of conservation of angular momentum and the fact that the force of gravitational attraction between a planet and the Sun acts along the line joining the centers of the two celestial objects.

Consider a cylindrical turntable whose mass is M and radius is R, turning with an initial angular speed (a) A parakeet of mass m, after hovering in flight above the outer edge of the turntable, gently lands on it and stays in one place on it, as shown in Figure 10-58. What is the angular speed of the turntable after the parakeet lands? (b) Becoming dizzy, the parakeet jumps off (not flies off) with a velocity relative to the turntable. The direction of is tangent to the edge of the turntable and in the direction of its rotation. What will be the angular speed of the turntable afterwards? Express your answer in terms of the two masses m and M, the radius R, the parakeet speed and the initial angular speed v1.

You are given a heavy but thin metal disk (like a coin, but larger; Figure 10-59) that has a mass of and a radius of (Objects like this are called Euler disks.) Placing the disk on a turntable, you spin the disk, on edge, about a vertical axis through a diameter of the disk and the center of the turntable. As you do this, you hold the turntable still with your other hand, letting it go immediately after you spin the disk. The turntable is a uniform solid cylinder with a radius equal to and a mass equal to and rotates on a frictionless bearing. The disk has an initial angular speed of (a) The disk spins down and falls over, finally coming to rest on the turntable with its symmetry axis coinciding with the turntable. What is the final angular speed of the turntable? (b) What will be the final angular speed if the disks symmetry axis ends up 0.100m from the axis of the turntable?

(a) Assuming Earth to be a homogeneous sphere that has a radius r and a mass m, show that the period T (time for one daily rotation) of Earths rotation about its axis is related to its radius by where Here L is the magnitude of the spin angular momentum of Earth. (b) Suppose that the radius r changes by a very small amount, due to some internal cause, such as thermal expansion. Show that the fractional change in the period is given approximately by (c) By how many kilometers would r need to increase for the period to change by (so that leap years would no longer be necessary)?

The term precession of the equinoxes refers to the fact that Earths spin axis does not stay fixed but sweeps out a cone once every (This explains why our pole star, Polaris, will not remain the pole star forever.) The reason for this instability is that Earth is a giant gyroscope. The spin axis of Earth precesses because of the torques exerted on it by the gravitational forces of the Sun and moon. The angle between the direction of Earths spin axis and the normal to the ecliptic plane (the plane of Earths orbit) is 22.5 degrees. Calculate an approximate value for this torque, given that the period of rotation of Earth is and its moment of inertia is

As indicated in the text, according to the Standard Model of Particle Physics, electrons are pointlike particles having no spatial extent. (This assumption has been confirmed experimentally, and the radius of the electron has been shown to be less than ) The intrinsic spin of an electron could in principle be due to its rotation. Let us check to see if this conclusion is feasible. (a) Assuming that the electron is a uniform sphere whose radius is what angular speed would be necessary to produce the observed intrinsic angular momentum of (b) Using this value of angular speed, show that the speed of a point on the equator of a spinning electron would be moving faster than the speed of light. What is your conclusion about the spin angular momentum being analogous to a spinning sphere with spatial extent?

An interesting phenomenon occurring in certain pulsars (see Problem 24) is an event known as a spin glitch, that is, a quick change in the spin rate of the pulsar due to a shift in mass location and a resulting rotational change in inertia. Imagine a pulsar whose radius is and whose period of rotation is The rotation period is observed to suddenly decrease from to If that decrease was caused by a contraction of the star, by what amount would the pulsar radius have had to change?

Figure 10-60 shows a pulley in the form of a uniform disk with a rope hanging over it. The circumference of the pulley is 1.2 m and its mass is 2.2 kg. The rope is 8.0 m long and its mass is 4.8 kg. At the instant shown in the figure, the system is at rest and the difference in height of the two ends of the rope is 0.60 m. (a) What is the angular speed of the pulley when the difference in height between the two ends of the rope is 7.2 m? (b) Obtain an expression for the angular momentum of the system as a function of time while neither end of the rope is above the center of the pulley. There is no slippage between rope and pulley wheel.