A day-care center has a merry-go-round that consists of a

Two points are on a disk that is turning about a fixed axis perpendicular to the disk and through its center at increasing angular velocity. One point is on the rim and the other point is halfway between the rim and the center. (a) Which point moves the greater distance in a given time? (b) Which point turns through the greater angle? (c) Which point has the greater speed? (d) Which point has the greater angular speed? (e) Which point has the greater tangential acceleration? ( f ) Which point has the greater angular acceleration? (g) Which point has the greater centripetal acceleration?

True or false: (a) Angular speed and linear speed have the same dimensions. (b) All parts of a wheel rotating about a fixed axis must have the same angular speed. (c) All parts of a wheel rotating about a fixed axis must have the same angular acceleration. (d) All parts of a wheel rotating about a fixed axis must have the same centripetal acceleration.

Starting from rest and rotating at constant angular acceleration, a disk takes 10 revolutions to reach an angular speed v. How many additional revolutions at the same angular acceleration are required for it to reach an angular speed of (a) (b) (c) (d) (e)

You are looking down from above at a merry-go-round and observe that it is rotating counterclockwise and that its rotation rate is slowing. If we designate counterclockwise as positive, what is the sign of the angular acceleration?

Chad and Tara go for a ride on a merry-go-round. Chad sits on a pony that is from the rotation axis, and Tara sits on a pony from the axis. The merry-go-round is traveling counterclockwise and is speeding up. Does Chad or Tara have (a) the larger linear speed? (b) the larger centripetal acceleration? (c) the larger tangential acceleration?

Disk B was identical to disk A before a hole was drilled though the center of disk B. Which disk has the largest moment of inertia about its symmetry axis center? Explain your answer.

CONTEXT-RICH The pitcher in a baseball game has a blazing fastball. You have not been able to swing the bat in time to hit the ball. You are now just trying to make the bat connect with the ball, hit the ball foul, and avoid a strikeout. So you decide to take your coachs advice and grip the bat high rather than at the very end. This change should increase bat speed; thus, you will be able to swing the bat quicker and increase your chances of hitting the ball. Explain how this theory works in terms of the moment of inertia, angular acceleration, and torque of the bat.

(a) Is the direction of an objects angular velocity necessarily the same as the direction net torque on it? Explain. (b) If net torque and angular velocity are in opposite directions, what does that tell you about the angular speed? (c) Can the angular velocity be zero even if the net torque is not zero? If your answer is yes, give an example.

A disk is free to rotate about a fixed axis. A tangential force applied a distance d from the axis causes an angular acceleration a. What angular acceleration is produced if the same force is applied a distance 2d from the axis? (a) (b) (c) (d) (e)

The moment of inertia of an object about an axis that does not pass through its center of mass is ________ the moment of inertia about a parallel axis through its center of mass: (a) always less than, (b) sometimes less than, (c) sometimes equal to, (d) always greater than.

The motor of a merry-go-round exerts a constant torque on it. As it speeds up from rest, the power output of the motor (a) is constant, (b) increases linearly with the angular speed of the merry-go-round, (c) is zero, (d) none of the above.

A constant net torque acts on a merry-go-round from startup until it reaches its operating speed. During this time, the merry-go-rounds kinetic energy (a) is constant, (b) increases linearly with angular speed, (c) increases quadratically as the square of the angular speed, (d) none of the above.

ENGINEERING APPLICATION Most doors knobs are designed so the knob is located on the side opposite the hinges (rather than in the center of the door, for example). Explain why this practice makes doors easier to open.

A wheel of radius R and angular speed v is rolling without slipping toward the north on a flat, stationary surface. The velocity of the point on the rim that is (momentarily) in contact with the surface is (a) equal in magnitude to Rv and directed toward the north, (b) equal to in magnitude Rv and directed toward the south, (c) zero, (d) equal to the speed of the center of mass and directed toward the north, (e) equal to the speed of the center of mass and directed toward the south.

A uniform solid cylinder and a uniform solid sphere have equal masses. Both roll on a horizontal surface without slipping. If their total kinetic energies are the same, then (a) the translational speed of the cylinder is greater than the translational speed of the sphere, (b) the translational speed of the cylinder is less than the translational speed of the sphere, (c) the translational speeds of the two objects are the same, (d), (a), (b), or (c) could be correct, depending on the radii of the objects.

Two identical-looking 1.0-m-long pipes are each plugged with of lead. In the first pipe, the lead is concentrated at the middle of the pipe, while in the second the lead is divided into two 5-kg masses placed at opposite ends of the pipe. The ends of the pipes are then sealed using four identical caps. Without opening either pipe, how could you determine which pipe has the lead at the ends?

Starting simultaneously from rest, a coin and a hoop roll without slipping down an incline. Which of the following statements is true? (a) The hoop reaches the bottom first. (b) The coin reaches the bottom first. (c) The coin and hoop arrive at the bottom simultaneously. (d) The race to the bottom depends on their relative masses. (e) The race to the bottom depends on their relative diameters.

For a hoop of mass M and radius R that is rolling without slipping, which is larger, its translational kinetic energy or its kinetic energy relative to the center of mass? (a) Its translational kinetic energy is larger. (b) Its kinetic energy relative to the center of mass is larger. (c) Both energies have the same magnitude. (d) The answer depends on the radius of the hoop. (e) The answer depends on the mass of the hoop.

For a disk of mass M and radius R that is rolling without slipping, which is larger, its translational kinetic energy or its kinetic energy relative to the center of mass? (a) Its translational kinetic energy is larger. (b) Its kinetic energy relative to the center of mass is larger. (c) Both energies have the same magnitude. (d) The answer depends on the radius of the disk. (e) The answer depends on the mass of the disk.

A perfectly rigid ball rolls without slipping along a perfectly rigid horizontal plane. Show that the frictional force acting on the ball must be zero. Hint: Consider a possible direction for the action of the frictional force and what effects such a force would have on the velocity of the center of mass and on the angular velocity.

A spool is free to rotate about a fixed axis, and a string wrapped around the axle of the spool causes the spool to rotate in a counterclockwise direction (Figure 9-42a). However, if the spool is set on a horizontal tabletop, the spool instead (given sufficient frictional force between the table and the spool) rotates in a clockwise direction and rolls to the right (Figure 9-42b). By considering torque about the appropriate axes, show that these conclusions are both consistent with Newtons second law for rotations.

You want to locate the center of gravity of an arbitrarily shaped flat object. You are told to suspend the object from a point, and to suspend a plumb line from the same point. Then draw a vertical line on the object to represent the plumb line. Next, you repeat the process using a different suspension point. The center of gravity will be at the intersection of the drawn lines. Explain the principle( s) behind this process.

A baseball is thrown at and with a spin rate of If the distance between the pitchers point of release and the catchers glove is about estimate how many revolutions the ball makes between release and catch. Neglect any effects of gravity or air resistance on the balls flight.

Consider the Crab Pulsar, discussed on page 293. Justify the statement that the loss in rotational energy is equivalent to the power output of The total power radiated by the Sun is about Assume that the pulsar has a mass that is , has a radius that is is rotating at about and has a rotational period that is increasing at

A 14-kg bicycle has 1.2-m-diameter wheels, each with a mass of The mass of the rider is Estimate the fraction of the total kinetic energy of the riderbicycle system that is associated with rotation of the wheels.

Why does toast falling off a table always land jelly-side down? The question may sound silly, but it has been a subject of serious scientific enquiry. The analysis is too complicated to reproduce here, but R. D. Edge and Darryl Steinert showed that a piece of toast, pushed gently over the edge of a table until it tilts off, typically falls off the table when it makes an angle of about with the horizontal (Figure 9-43) and at that instant has an angular speed of where is the length of one edge of the piece of toast (assumed to be square).* Assuming that a piece of toast is jelly-side up, what side will it land on if it falls from a 0.500-m-high table? If it falls from a 1.00-m-high table? Assume that _ _ 10.0 cm. Ignore any forces due to air resistance.

Consider your moment of inertia about a vertical axis through the center of your body, both when you are standing straight up with your arms flat against your sides, and when you are standing straight up holding your arms straight out to your sides. Estimate the ratio of the moment of inertia with your arms straight out to the moment of inertia with your arms flat against your sides.

A particle moves with a constant speed of in a 90-m-radius circle. (a) What is its angular speed in radians per second about the center of the circle? (b) How many revolutions does it make in

Awheel released from rest is rotating with constant angular acceleration of At after the release: (a) What is its angular speed? (b) Through what angle has the wheel turned? (c) How many revolutions has it completed? (d) What is the linear speed and what is the magnitude of the linear acceleration of a point from the axis of rotation?

MULTISTEP When a turntable rotating at is shut off, it comes to rest in Assuming constant angular acceleration, find (a) the angular acceleration. During the find (b) the average angular speed and (c) the angular displacement in revolutions.

A12-cm-radius disk that begins to rotate about its axis at rotates with a constant angular acceleration of At (a) what is the angular speed of the disk, and (b) what are the tangential and centripetal components of the acceleration of a point on the edge of the disk?

A12-m-radius Ferris wheel rotates once each (a) What is its angular speed (in radians per second)? (b) What is the linear speed of a passenger? (c) What is the acceleration of a passenger?

A cyclist accelerates uniformly from rest. After the wheels have rotated (a) What is the angular acceleration of the wheels? (b) What is the angular speed of the wheels at the end of the

What is the angular speed of Earth in radians per second as it rotates about its axis?

Awheel rotates through in as it is brought to rest with constant angular acceleration. Determine the wheels initial angular speed before braking began.

A bicycle has 0.750-m-diameter wheels. The bicyclist accelerates from rest with constant acceleration to in What is the angular acceleration of the wheels?

ENGINEERING APPLICATION The tape in a standard VHS videotape cassette has a total length of which is enough for the tape to play for (Figure 9-44). As the tape starts, the full reel has a 45-mm outer radius and a 12-mm inner radius. At some point during the play, both reels have the same angular speed. Calculate this angular speed in radians per second and in revolutions per minute. Hint: Between the two reels the tape moves at constant speed. SSM

CONTEXT-RICH To start a lawn mower, you must pull on a rope wound around the perimeter of a flywheel. After you pull the rope for the flywheel is rotating at 4.5 revolutions per second, at which point the rope disengages. This attempt at starting the mower does not work, however, and the flywheel slows, coming to rest after the disengagement. Assume constant acceleration during both spin-up and spin-down. (a) Determine the average angular acceleration during the 4.5-s spin-up and again during the 0.24-s spin-down. (b) What is the maximum angular speed reached by the flywheel? (c) Determine the ratio of the number of revolutions made during spin-up to the number made during spin-down.

Mars orbits the Sun at a mean orbital radius of and has an orbital period of Earth orbits the Sun at a mean orbital radius of (a) The EarthSun line sweeps out an angle of during one Earthyear. Approximately what angle is swept out by the Mars-Sun line during one Earthyear? (b) How frequently are Mars and the Sun in opposition (on diametrically opposite sides of Earth)?

A tennis ball has a mass of and a diameter of Find the moment of inertia about its diameter. Model the ball as a thin spherical shell.

Four particles, one at each of the four corners of a square with 2.0-m-long edges, are connected by massless rods (Figure 9-45). The masses of the particles are and Find the moment of inertia of the system about the z axis. SSM

Use the parallel-axis theorem and the result for Problem 41 to find the moment of inertia of the four-particle system in Figure 9-45 about an axis that passes through the center of mass and is parallel with the z axis. Check your result by direct computation.

For the four-particle system of Figure 9-45, (a) find the moment of inertia about the x axis, which passes through and and (b) find the moment of inertia about the y axis, which passes through and

Determine the moment of inertia of a uniform solid sphere of mass M and radius R about an axis that is tangent to the surface of the sphere (Figure 9-46).

A 1.00-m-diameter wagon wheel consists of a thin rim having a mass of and each with a mass of Determine the moment of inertia of the wagon wheel about its axis.

MULTISTEP Two point masses and are separated by a massless rod of length L. (a) Write an expression for the moment of inertia I about an axis perpendicular to the rod and passing through it a distance x from mass (b) Calculate and show that I is at a minimum when the axis passes through the center of mass of the system.

A uniform rectangular plate has mass m and edges of lengths a and b. (a) Show by integration that the moment of inertia of the plate about an axis that is perpendicular to the plate and passes through one corner is (b) What is the moment of inertia about an axis that is perpendicular to the plate and passes through its center of mass?

In attempting to ensure a spot on the pep squad, you and your friend Corey research baton-twirling. Each of you is using “The Beast” as a model baton: two uniform spheres, each of mass and radius mounted at the ends of a 30.0-cm uniform rod of mass (Figure 9-47). You want to determine the moment of inertia I of “The Beast” about an axis perpendicular to the rod and passing through its center. Corey uses the approximation that the two spheres can be treated as point particles that are from the axis of rotation, and that the mass of the rod is negligible. You, however, decide to do an exact calculation. (a) Compare the two results. (Give the percentage difference between them). (b) Suppose the spheres were replaced by two thin spherical shells, each of the same mass as the original solid spheres. Give a conceptual argument explaining how this replacement would, or would not, change the value of I.

The methane molecule has four hydrogen atoms located at the vertices of a regular tetrahedron of edge length with the carbon atom at the center of the tetrahedron (Figure 9-48). Find the moment of inertia of this molecule for rotation about an axis that passes through the centers of the carbon atom and one of the hydrogen atoms.

A hollow cylinder has mass m, an outside radius and an inside radius Use integration to show that the moment of inertia about its axis is given by Hint: Review Section 9-3, where the moment of inertia is calculated for a solid cylinder by direct integration.

BIOLOGICAL APPLICATION While slapping the waters surface with his tail to communicate danger, a beaver must rotate it about one of its narrow ends. Let us model the tail as a rectangle of uniform thickness and density (Figure 9-49). Estimate its moment of inertia about the line passing through its narrow end (dashed line). Assume that the tail measures 15 by with a thickness of 1.0 cm and that the flesh has the density of water.

CONTEXT-RICH To prevent damage to her shoulders, your elderly grandmother wants to purchase the rug beater (Figure 9-50) with the lowest moment of inertia about its grip end. Knowing you are taking physics, she asks your advice. There are two models to choose from. Model A has a 1.0-m-long handle on a 40-cm-edge-length square, where the masses of the handle and square are and respectively. Model B has a 0.75-m-long handle and a 30-cmedge- length square, where the masses of the handle and square are and respectively. Which model should you recommend? Determine which beater is easier to swing from the very end by computing the moment of inertia for both beaters.

Use integration to show that the moment of inertia of a thin spherical shell of radius R and mass m about an axis through its center is

According to one model, the density of Earth varies with the distance r from the center of Earth as where R is the radius of Earth and C is a constant. (a) Find C in terms of the total mass M and the radius R. (b) According to this model, what is the moment of inertia of Earth about an axis through its center. (See Problem 53.)

Use integration to determine the moment of inertia about its axis of a uniform right circular cone of height H, base radius R, and mass M.

Use integration to determine the moment of inertia of a thin uniform disk of mass Mand radius R about an axis in the plane of the disk and passing through its center. Check your answer by referring to Table 9-1.

ENGINEERING APPLICATION, CONTEXT-RICH An advertising firm has contacted your engineering firm to create a new advertisement for a local ice-cream stand. The owner of this stand wants to add rotating solid cones (painted to look like ice-cream cones, of course) to catch the eye of travelers. Each cone will rotate about an axis parallel to its base and passing through its apex. The actual size of the cones is to be decided upon, and the owner wonders if it would be more energy-efficient to rotate smaller cones than larger ones. He asks your firm to write a report showing the determination of the moment of inertia of a homogeneous right circular cone of height H, base radius R, and mass M. What is the result of your report?

ENGINEERING APPLICATION, CONTEXT-RICH A firm wants to determine the amount of frictional torque in their current line of grindstones, so they can redesign them to be more energy efficient. To do this, they ask you to test the best-selling model, which is basically a disk-shaped grindstone of mass and radius that operates at When the power is shut off, you time the grindstone and find it takes for it to stop rotating. (a) What is the angular acceleration of the grindstone? (Assume constant angular acceleration.) (b) What is the frictional torque exerted on the grindstone?

A 2.5-kg 11-cm-radius cylinder, initially at rest, is free to rotate about the axis of the cylinder. Arope of negligible mass is wrapped around it and pulled with a force of Assuming that the rope does not slip, find (a) the torque exerted on the cylinder by the rope, (b) the angular acceleration of the cylinder, and (c) the angular speed of the cylinder after

A grinding wheel is initially at rest. A constant external torque of is applied to the wheel for giving the wheel an angular speed of The external torque is then removed, and the wheel comes to rest later. Find (a) the moment of inertia of the wheel, and (b) the frictional torque, which is assumed to be constant.

Apendulum consisting of a string of length L attached to a bob of mass m swings in a vertical plane. When the string is at an angle u to the vertical, (a) calculate the tangential acceleration of the bob using (b) What is the torque exerted about the pivot point? (c) Show that with gives the same tangential acceleration as found in Part (a).

A uniform rod of mass M and length L is pivoted at one end and hangs as in Figure 9-51 so that it is free to rotate without friction about its pivot. It is struck a sharp horizontal blow a distance x below the pivot, as shown. (a) Show that, just after the rod is struck, the speed of the center of mass of the rod is given by where and are the average force and duration, respectively, of the blow. (b) Find the horizontal component of the force exerted by the pivot on the rod, and show that this force component is zero if This point (the point of impact when the horizontal component of the pivot force is zero) is called the center of percussion of the rod-pivot system.

MULTISTEP Auniform horizontal disk of mass Mand radius R is spinning about the vertical axis through its center with an angular speed v. When the spinning disk is dropped onto a horizontal tabletop, kinetic-frictional forces on the disk oppose its spinning motion. Let be the coefficient of kinetic friction between the disk and the tabletop. (a) Find the torque dt exerted by the force of friction on a circular element of radius r and width dr. (b) Find the total torque exerted by friction on the disk. (c) Find the time required for the disk to stop rotating.

The particles in Figure 9-52 are connected by a very light rod. They rotate about the y axis at (a) Find the speed of each particle, and use it to calculate the kinetic energy of this system directly from (b) Find the moment of inertia about the y axis, calculate the kinetic energy from and compare your result with your Part-(a) result.

A 1.4-kg 15-cm-diameter solid sphere is rotating about its diameter at \(70 \mathrm{rev} / \mathrm{min}\). (a) What is its kinetic energy? (b) If an additional \(5.0 \mathrm{~mJ}\) of energy are added to the kinetic energy, what is the new angular speed of the sphere?

Calculate the kinetic energy of Earth due to its spinning about its axis, and compare your answer with the kinetic energy of the orbital motion of Earths center of mass about the Sun. Assume Earth to be a homogeneous sphere of mass and radius The radius of Earths orbit is

A 2000-kg block is lifted at a constant speed of by a steel cable that passes over a massless pulley to a motor-driven winch (Figure 9-53). The radius of the winch drum is (a) What is the tension in the cable? (b) What torque does the cable exert on the winch drum? (c) What is the angular speed of the winch drum? (d) What power must be developed by the motor to drive the winch drum? SSM

A uniform disk that has a mass M and a radius R can rotate freely about a fixed horizontal axis that passes through its center and is perpendicular to the plane of the disk. A small particle that has a mass m is attached to the rim of the disk at the top, directly above the pivot. The system is gently nudged, and the disk begins to rotate. As the particle passes through its lowest point, (a) what is the angular speed of the disk, and (b) what force is exerted by the disk on the particle?

Auniform 1.5-m-diameter ring is pivoted at a point on its perimeter so that it is free to rotate about a horizontal axis that is perpendicular to the plane of the ring. The ring is released with the center of the ring at the same height as the axis (Figure 9-54). (a) If the ring was released from rest, what was its maximum angular speed? (b) What minimum angular speed must it be given at release if it is to rotate a full 360?

ENGINEERING APPLICATION, CONTEXT-RICH You set out to design a car that uses the energy stored in a flywheel consisting of a uniform 100-kg cylinder of radius R that has a maximum angular speed of The flywheel must deliver an average of of energy for each kilometer of distance. Find the smallest value of R for which the car can travel without the flywheel needing to be reenergized.

The system shown in Figure 9-55 consists of a 4.0-kg block resting on a frictionless horizontal ledge. This block is attached to a string that passes over a pulley, and the other end of the string is attached to a hanging 2.0-kg block. The pulley is a uniform disk of radius and mass Find the acceleration of each block and the tension in the string.

For the system in Problem 71, the 2.0-kg block is released from rest. (a) Find the speed of the block after it falls a distance of (b) What is the angular speed of the pulley at this instant?

For the system in Problem 71, if the (frictionless) ledge were adjustable in angle, at what angle would it have to be tilted upward so that once the system is set into motion the blocks will continue to move at constant speed?

In the system shown in Figure 9-55, there is a 4.0-kg block resting on a horizontal ledge. The coefficient of kinetic friction between the ledge and the block is 0.25. The block is attached to a string that passes over a pulley, and the other end of the string is attached to a hanging 2.0-kg block. The pulley is a uniform disk of radius 8.0 cm and mass Find the acceleration of each block and the tensions in the segments of string between each block and the pulley.

A 1200-kg car is being raised over water by a winch. At the moment the car is above the water (Figure 9-56), the gearbox breaks, allowing the winch drum to spin freely as the car falls. During the cars fall, there is no slipping between the (massless) rope, the pulley wheel, and the winch drum. The moment of inertia of the winch drum is and the moment of inertia of the pulley wheel is The radius of the winch drum is and the radius of the pulley is Assume that the car starts to fall from rest. Find the speed of the car as it hits the water.

The system in Figure 9-57 is released from rest when the 30-kg block is above the ledge. The pulley is a uniform 5.0-kg disk with a radius of Just before the 30-kg block hits the ledge, find (a) its speed, (b) the angular speed of the pulley, and (c) the tensions in the strings. (d) Find the time of descent for the 30-kg block. Assume that the string does not slip on the pulley.

A uniform solid sphere of mass M and radius R is free to rotate about a horizontal axis through its center. A string is wrapped around the sphere and is attached to an object of mass m (Figure 9-58). Assume that the string does not slip on the sphere. Find (a) the acceleration of the object and (b) the tension in the string.

Two objects, of masses and are connected by a string of negligible mass that passes over a pulley with frictionless bearings (Figure 9-59). The pulley is a uniform 50.0-g disk with a radius of The string does not slip on the pulley. (a) Find the accelerations of the objects. (b) What is the tension in the string between the 500-g block and the pulley? What is the tension in the string between the 510-g block and the pulley? By how much do these tensions differ? (c) What would your answers be if you neglected the mass of the pulley?

Two objects are attached to ropes that are attached to two wheels on a common axle, as shown in Figure 9-60. The two wheels are attached together so that they form a single rigid object. The moment of inertia of the rigid object is The radii of the wheels are and (a) If find such that there is no angular acceleration of the wheels. (b) If is placed on top of find the angular acceleration of the wheels and the tensions in the ropes.

The upper end of the string wrapped around the cylinder in Figure 9-61 is held by a hand that is accelerated upward so that the center of mass of the cylinder does not move as the cylinder spins up. Find (a) the tension in the string, (b) the angular acceleration of the cylinder, and (c) the acceleration of the hand.

A uniform cylinder of mass and radius R is pivoted on frictionless bearings. Amassless string wrapped around the cylinder is connected to a block of mass that is on a frictionless incline of angle u as shown in Figure 9-62. The system is released from rest when the block is at a vertical distance h above the bottom of the incline. (a) What is the acceleration of the block? (b) What is the tension in the string? (c) What is the speed of the block as it reaches the bottom of the incline? (d) Evaluate your answers for the special case where and Are your answers what you would expect for this special case? Explain. SSM

A device for measuring the moment of inertia of an object is shown in Figure 9-63. The circular platform is attached to a concentric drum of radius R, and the platform and the drum are free to rotate about a frictionless vertical axis. The string that is wound around the drum passes over a frictionless and massless pulley to a block of mass M. The block is released from rest, and the time t it takes for it to drop a distance D is measured. The system is then rewound, the object whose moment of inertia I we wish to measure is placed on the platform, and the system is again released from rest. The time required for the block to drop the same distance D then provides the data needed to calculate I. Using and (a) find the moment of inertia of the platformdrum combination. (b) Find the moment of inertia of the platformdrum- object combination. (c) Use your results for Parts (a) and (b) to find the moment of inertia of the object.

A homogeneous 60-kg cylinder of radius is rolling without slipping along a horizontal floor at a speed of What is the minimum amount of work that was required to give it this motion?

An object is rolling without slipping. What percentage of its total kinetic energy is its translational kinetic energy if the object is (a) a uniform sphere, (b) a uniform cylinder, or (c) a hoop?

In 1993 a giant 400-kg yo-yo with a radius of was dropped from a crane at height of One end of the string was tied to the top of the crane, so the yo-yo unwound as it descended. Assuming that the axle of the yo-yo had a radius of estimate its linear speed at the end of the fall.

A uniform cylinder of mass M and radius R has a string wrapped around it. The string is held fixed, and the cylinder falls vertically as shown in Figure 9-61. (a) Show that the acceleration of the cylinder is downward with a magnitude (b) Find the tension in the string.

A 0.10-kg yo-yo consisting of two solid disks, each of radius is joined by a massless rod of radius A string is wrapped around the rod. One end of the string is held fixed and is under tension as the yo-yo is released. The yo-yo rotates as it descends vertically. Find (a) the acceleration of the yo-yo, and (b) the tension T.

Auniform solid sphere rolls down an incline without slipping. If the linear acceleration of the center of mass of the sphere is 0.20g, then what is the angle the incline makes with the horizontal?

A thin spherical shell rolls down an incline without slipping. If the linear acceleration of the center of mass of the shell is 0.20g, what is the angle the incline makes with the horizontal?

A basketball rolls without slipping down an incline of angle u. The coefficient of static friction is ms Model the ball as a thin spherical shell. Find (a) the acceleration of the center of mass of the ball, (b) the frictional force acting on the ball, and (c) the maximum angle of the incline for which the ball will roll without slipping.

A uniform solid cylinder of wood rolls without slipping down an incline of angle u. The coefficient of static friction is Find (a) the acceleration of the center of mass of the cylinder, (b) the frictional force acting on the cylinder, and (c) the maximum angle of the incline for which the cylinder will roll without slipping.

Released from rest at the same height, a thin spherical shell and solid sphere of the same mass m and radius R roll without slipping down an incline through the same vertical drop H (Figure 9-64). Each is moving horizontally as it leaves the ramp. The spherical shell hits the ground a horizontal distance L from the end of the ramp and the solid sphere hits the ground a distance from the end of the ramp. Find the ratio L_ >L.

A uniform, thin cylindrical shell and a solid cylinder roll horizontally without slipping. The speed of the cylindrical shell is v. The solid cylinder and the hollow cylinder encounter an incline that they climb without slipping. If the maximum height they reach is the same, find the initial speed of the solid cylinder.

A thin cylindrical shell and a solid sphere start from rest and roll without slipping down a 3.0-m-long inclined plane. The cylinder arrives at the bottom of the incline after the sphere. Determine the angle the incline makes with the horizontal.

A wheel has a thin 3.0-kg rim and four spokes, each of mass 1.2 kg. Find the kinetic energy of the wheel when it is rolling at 6.0 m/s on a horizontal surface.

A uniform solid cylinder of mass M and radius R is at rest on a slab of mass m, which in turn rests on a horizontal, frictionless table (Figure 9-65). If a horizontal force \(\vec{F}\) is applied to the slab, it accelerates and the cylinder rolls without slipping. Find the acceleration of the slab in terms of M, R, and F.

(a) Find the angular acceleration of the cylinder in Problem 96. Is the cylinder rotating clockwise or counterclockwise? (b) What is the cylinders linear acceleration (magnitude and direction) relative to the table? (c) What is the magnitude and direction of the linear acceleration of the center of mass of the cylinder relative to the slab?

MULTISTEP If the force in Problem 96 acts over a distance d, in terms of the symbols given, find (a) the kinetic energy of the slab, and (b) the total kinetic energy of the cylinder. (c) Show that the total kinetic energy of the slab-cylinder system is equal to the work done by the force.

ENGINEERING APPLICATION Two large gears that are being designed as part of a large machine are shown in Figure 9-66; each is free to rotate about a fixed axis through its center. The radius and moment of inertia of the smaller gear are and respectively, and the radius and moment of inertia of the larger gear are and respectively. The lever attached to the smaller gear is long and has a negligible mass. (a) If a worker will typically apply a force of to the end of the lever, as shown, what will be the angular accelerations of the two gears? (b) Another part of the machine (not shown) will apply a force tangentially to the outer edge of the larger gear to temporarily keep the gear system from rotating. What should the magnitude and direction of this force (clockwise or counterclockwise) be? SSM

ENGINEERING APPLICATION, CONTEXT-RICH As the chief design engineer for a major toy company, you are in charge of designing a loop-the-loop toy for youngsters. The idea, as shown in Figure 9-67, is that a ball of mass m and radius r will roll down an inclined track and around the loop without slipping. The ball starts from rest at a height h above the tabletop that supports the whole track. The loop radius is R. Determine the minimum height h, in terms of R and r, for which the ball will remain in contact with the track during the whole of its loop-the-loop journey. (Do not neglect the size of the balls radius when doing this calculation.)

Abowling ball of mass Mand radius R is released so that at the instant it touches the floor it is moving horizontally with a speed and is not rotating. It slides for a time a distance s1 before it begins to roll without slipping. (a) If is the coefficient of kinetic friction between the ball and the floor, find and the final speed of the ball. (b) Find the ratio of the final kinetic energy to the initial kinetic energy of the ball. (c) Evaluate and for and

CONTEXT-RICH During a game of pool, the cue ball (a uniform sphere of radius r) is at rest on the horizontal pool table (Figure 9-68). You strike the ball horizontally with your cue stick, which delivers a large horizontal force of magnitude for a short time. The stick strikes the ball at a point a vertical height h above the tabletop. Assume that the striking location is above the balls center. Show that the balls angular speed v is related to the initial linear speed of its center of mass by Estimate the balls rotation rate just after the hit using reasonable estimates for h, r, and vc

A uniform solid sphere is set rotating about a horizontal axis at an angular speed and then is placed on the floor with its center of mass at rest. If the coefficient of kinetic friction between the sphere and the floor is find the speed of the center of mass of the sphere when the sphere begins to roll without slipping.

A uniform solid ball that has a mass of and a radius of rests on a horizontal surface. A sharp force is applied to the ball in a horizontal direction above the horizontal surface. During impact the force increases linearly from to in and then it decreases linearly to in (a) What is the speed of the ball just after impact? (b) What is the angular speed of the ball after impact? (c) What is the speed of the ball when it begins to roll without slipping? (d) How far does the ball travel along the surface before it begins to roll without slipping? Assume that

A 0.16-kg billiard ball whose radius is is given a sharp blow by a cue stick. The applied force is horizontal and the line of action of the force passes through the center of the ball. The speed of the ball just after the blow is and the coefficient of kinetic friction between the ball and the billiard table is 0.60. (a) How long does the ball slide before it begins to roll without slipping? (b) How far does it slide? (c) What is its speed once it begins rolling without slipping?

A billiard ball that is initially at rest is given a sharp blow by a cue stick. The force is horizontal and is applied at a distance below the centerline, as shown in Figure 9-69. The speed of the ball just after the blow is and the coefficient of kinetic friction between the ball and the billiard table is (a) What is the angular speed of the ball just after the blow? (b) What is the speed of the ball once it begins to roll without slipping? (c) What is the kinetic energy of the ball just after the hit?

A bowling ball of radius R has an initial speed down the lane and a forward spin just after its release. The coefficient of kinetic friction is (a) What is the speed of the ball just as it begins rolling without slipping? (b) For how long a time does the ball slide before it begins rolling without slipping? (c) What distance does the ball slide down the lane before it begins rolling without slipping?

The radius of a small playground merry-go-round is To start it rotating, you wrap a rope around its perimeter and pull with a force of for During this time, the merry-goround makes one complete rotation. Neglect any effects of friction. (a) Find the angular acceleration of the merry-go-round. (b) What torque is exerted by the rope on the merry-go-round? (c) What is the moment of inertia of the merry-go-round?

Auniform 2.00-m-long stick is raised at an angle of to the horizontal above a sheet of ice. The bottom end of the stick rests on the ice. The stick is released from rest. The bottom end of the stick remains in contact with the ice at all times. How far will the bottom end of the stick have traveled during the time the rest of the stick is falling to the ice? Assume that the ice is frictionless.

Auniform 5.0-kg disk has a radius of and is pivoted so that it rotates freely about its axis (Figure 9-70). A string wrapped around the disk is pulled with a force equal to (a) What is the torque being exerted by this force about the rotation axis? (b) What is the angular acceleration of the disk? (c) If the disk starts from rest, what is its angular speed after (d) What is its kinetic energy afterthe (e) What is the angular displacement of the disk during the (f ) Show that the work done by the torque, equals the kinetic energy.

A uniform 0.25-kg thin rod that has an length is free to rotate about a fixed horizontal axis perpendicular to and through one end of the rod. It is held horizontal and released. Immediately after it is released, what is (a) the acceleration of the center of the rod, and (b) the initial acceleration of the free end of the rod? (c) What is the speed of the center of mass of the rod when the rod is (momentarily) vertical?

A marble of mass M and radius R rolls without slipping down the track on the left from a height as shown in Figure 9-71. The marble then goes up the frictionless track on the right to a height h Find h2

A uniform 120-kg disk with a radius equal to initially rotates with an angular speed of A constant tangential force is applied at a radial distance of from the axis. (a) How much work must this force do to stop the wheel? (b) If the wheel is brought to rest in what torque does the force produce? What is the magnitude of the force? (c) How many revolutions does the wheel make in these

A day-care center has a merry-go-round that consists of a uniform 240-kg circular wooden platform in diameter. Four children run alongside the merry-go-round and push tangentially along the platforms circumference until, starting from rest, the merry-go-round is spinning at During the spin-up: (a) If each child exerts a sustained force equal to how far does each child run? (b) What is the angular acceleration of the merry-go-round? (c) How much work does each child do? (d) What is the increase in the kinetic energy of the merry-go-round?

A uniform 1.5-kg hoop with a 65-cm radius has a string wrapped around its circumference and lies flat on a horizontal frictionless table. The free end of the string is pulled with a constant horizontal force equal to and the string does not slip on the hoop. (a) How far does the center of the hoop travel in (b) What is the angular speed of the hoop after

A hand-driven grinding wheel is a uniform 60-kg disk with a 45-cm radius. It has a handle of negligible mass from the rotation axis. A compact 25-kg load is attached to the handle when it is at the same height as the horizontal rotation axis. Ignoring the effects of friction, find (a) the initial angular acceleration of the wheel, and (b) the maximum angular speed of the wheel.

A uniform disk of radius R and mass M is pivoted about a horizontal axis parallel to its symmetry axis and passing through a point on its perimeter, so that it can swing freely in a vertical plane (Figure 9-72). It is released from rest with its center of mass at the same height as the pivot. (a) What is the angular speed of the disk when its center of mass is directly below the pivot? (b) What force is exerted by the pivot on the disk at this moment?

ENGINEERING APPLICATION The roof of the student dining hall at your college will be supported by high cross-braced wooden beams attached in the shape of an upside-down L (Figure 9-73). Each vertical beam is high and wide, and the horizontal cross-member is long. The mass of the vertical beam is and the mass of the horizontal beam is As the workers were building the hall, one of these structures started to fall over before it was anchored into place. (Luckily they stopped it before it fell.) (a) If it started falling from an upright position, what was the initial angular acceleration of the structure? Assume that the bottom did not slide across the floor and that it did not fall out of plane; that is, during the fall, the structure remained in the vertical plane defined by the initial position of the structure. (b) What would be the magnitude of the initial linear acceleration of the upper right corner of the horizontal beam? (c) What would the horizontal component of the initial linear acceleration be at this same location? (d) Estimate the structures rotational speed at impact.

CONTEXT-RICH You are participating in league bowling with your friends. Time after time, you notice that your bowling ball rolls back to you without slipping on the flat section of track. When the ball encounters the slope that brings it up to the ball return, it is moving at The length of the sloped part of the track is 2.50m. The radius of the bowling ball is 11.5 cm. (a) What is the angular speed of the ball before it encounters the slope? (b) If the speed with which the ball emerges at the top of the incline is what is the angle (assumed constant) that the sloped section of the track makes with the horizontal? (c) What is the magnitude of the angular acceleration of the ball while it is on the slope?

Figure 9-74 shows a hollow cylinder that has a length equal to a mass equal to and radius equal to The cylinder is free to rotate about a vertical axis that passes through its center and is perpendicular to the cylinder. Two objects are inside the cylinder. Each object has a mass equal to and is attached to a spring that has a force constant k and an unstressed length equal to The inside walls of the cylinder are frictionless. (a) Determine the value of the force constant if the objects are located from the center of the cylinder when the cylinder rotates at (b) How much work is required to bring the system from rest to an angular speed of 24 rad>s?

Apopular classroom demonstration involves taking a meterstick and holding it horizontally at the 0.0-cm end with a number of pennies spaced evenly along its surface. If the hand is suddenly relaxed so that the meterstick pivots freely about the 0.0-cm mark under the influence of gravity, an interesting thing is seen during the first part of the sticks rotation: the pennies nearest the 0.0-cm mark remain on the meterstick, while those nearest the 100-cm mark are left behind by the falling meterstick. (This demonstration is often called the faster than gravity demonstration.) Suppose this demonstration is repeated without any pennies on the meterstick. (a) What would the initial acceleration of the 100.0-cm mark then be? (The initial acceleration is the acceleration just after the release.) (b) What point on the meterstick would then have an initial acceleration greater than g?

A solid metal rod long is free to pivot without friction about a fixed horizontal axis perpendicular to the rod and passing through one of its ends. The rod is held in a horizontal position. Small coins, each of mass m, are placed on the rod from the pivot. If the free end is now released, calculate the initial force exerted on each coin by the rod. Assume that the masses of the coins can be ignored in comparison to the mass of the rod.

Suppose that for the system described in Problem 120, the force constants are each The system starts from rest and slowly accelerates until the masses are from the center of the cylinder. How much work was done in the process?

A string is wrapped around a uniform solid cylinder of radius R and mass M that rests on a horizontal frictionless surface. (The string does not touch the surface because there is a groove cut in the surface to provide space for the string to clear.) The string is pulled horizontally from the top with force F. (a) Show that the magnitude of the angular acceleration of the cylinder is twice the magnitude of the angular acceleration needed for rolling without slipping, so that the bottom point on the cylinder slides backward against the table. (b) Find the magnitude and direction of the frictional force between the table and cylinder that would be needed for the cylinder to roll without slipping. What would be the magnitude of acceleration of the cylinder in this case?

SPREADSHEET Let us calculate the position y of the falling load attached to the winch in Example 9-8 as a function of time by numerical integration. Let the direction be straight downward. Then or where t is the time taken for the bucket to fall a distance is a small increment of and Hence, we can calculate t as a function of d by numerical summation. Make a graph of y versus t between and Assume that and Use Compare this position to the position of the falling load if it were in free-fall.

Figure 9-75 shows a solid cylinder that has mass M and radius R to which a second solid cylinder that has mass m and radius r is attached. Astring is wound about the smaller cylinder. The larger cylinder rests on a horizontal surface. The coefficient of static friction between the larger cylinder and the surface is If a light tension is applied to the string in the vertical direction, the cylinder will roll to the left; if the tension is applied with the string horizontally to the right, the cylinder rolls to the right. Find the angle between the string and the horizontal that will allow the cylinder to remain stationary when a light tension is applied to the string.

In problems dealing with a pulley with a nonzero moment of inertia, the magnitude of the tensions in the ropes hanging on either side of the pulley are not equal. The difference in the tension is due to the static frictional force between the rope and the pulley; however, the static frictional force cannot be made arbitrarily large. Consider a massless rope wrapped partly around a cylinder through an angle \(\Delta \theta\) (measured in radians). It can be shown that if the tension on one side of the pulley is T, while the tension on the other side is \(T^{\prime}\left(T^{\prime}>T\right)\) the maximum value of \(T^{\prime}\) that can be maintained without the rope slipping is \(T_{\max }^{\prime}=T e^{\mu_{s} \Delta \theta}\) where \(\mu_{\mathrm{s}}\) is the coefficient of static friction. Consider the Atwood’s machine in Figure 9-76: the pulley has a radius the moment of inertia is \(I=0.35 \mathrm{\ kg} \cdot \mathrm{m}^{2}\) and the coefficient of static friction between the wheel and the string is \(\mu_{\mathrm{s}}=0.30\) (a) If the tension on one side of the pulley is what is the maximum tension on the other side that will prevent the rope from slipping on the pulley? (b) What is the acceleration of the blocks in this case? (c) If the mass of one of the hanging blocks is what is the maximum mass of the other block if, after the blocks are released, the pulley is to rotate without slipping?

A massive, uniform cylinder has a mass m and a radius R (Figure 9-77). It is accelerated by a tension force that is applied through a rope wound around a light drum of radius r that is attached to the cylinder. The coefficient of static friction is sufficient for the cylinder to roll without slipping. (a) Find the frictional force. (b) Find the acceleration a of the center of the cylinder. (c) Show that it is possible to choose r so that a is greater than (d) What is the direction of the frictional force in the circumstances of Part (c)?

Auniform rod that has a length L and a mass Mis free to rotate about a horizontal axis through one end, as shown in Figure 9-78. The rod is released from rest at Show that the parallel and perpendicular components of the force exerted by the axis on the rod are given by and where is the component parallel with the rod and is the component perpendicular to the rod.