Answer: A solid, horizontal cylinder of mass 10.0 kg and

What is the magnitude of the angular acceleration of a 25.0-kg disk of radius 0.800 m when a torque of magnitude 40.0 N ? m is applied to it? (a) 2.50 rad/s2 (b) 5.00 rad/s2 (c) 7.50 rad/s2 (d) 10.0 rad/s2 (e) 12.5 rad/s2

A horizontal plank 4.00 m long and having mass 20.0 kg rests on two pivots, one at the left end and a second 1.00 m from the right end. Find the magnitude of the force exerted on the plank by the second pivot. (a) 32.0 N (b) 45.2 N (c) 112 N (d) 131 N (e) 98.2 N

A wrench 0.500 m long is applied to a nut with a force of 80.0 N. Because of the cramped space, the force must be exerted upward at an angle of 60.0 with respect to a line from the bolt through the end of the wrench. How much torque is applied to the nut? (a) 34.6 N ? m (b) 4.56 N ? m (c) 11.8 N ? m (d) 14.2 N ? m (e) 20.0 N ? m

As shown in Figure MCQ8.4, a cord is wrapped onto a cylindrical reel mounted on a fixed, frictionless, horizontal axle. When does the reel have a greater magnitude of angular acceleration? (a) When the cord is pulled down with a constant force of 50 N. (b) When an object of weight 50 N is hung from the cord and released. (c) The angular accelerations in (a) and (b) are equal. (d) It is impossible to determine.

Two forces are acting on an object. Which of the following statements is correct? (a) The object is in equilibrium if the forces are equal in magnitude and opposite in direction. (b) The object is in equilibrium if the net torque on the object is zero. (c) The object is in equilibrium if the forces act at the same point on the object. (d) The object is in equilibrium if the net force and the net torque on the object are both zero. (e) The object cannot be in equilibrium because more than one force acts on it.

A block slides down a frictionless ramp, while a hollow sphere and a solid ball roll without slipping down a second ramp with the same height and slope. Rank the arrival times at the bottom from shortest to longest. (a) sphere, ball, block (b) ball, block, sphere (c) ball, sphere, block (d) block, sphere, ball (e) block, ball, sphere

A constant net nonzero torque is exerted on an object. Which of the following quantities cannot be constant for this object? More than one answer may be correct. (a) angular acceleration (b) angular velocity (c) moment of inertia (d) center of mass (e) angular momentum

A disk rotates about a fixed axis that is perpendicular to the disk and passes through its center. At any instant, does every point on the disk have the same (a) centripetal acceleration, (b) angular velocity, (c) tangential acceleration, (d) linear velocity, or (e) total acceleration?

A solid disk and a hoop are simultaneously released from rest at the top of an incline and roll down without slipping. Which object reaches the bottom first? (a) The one that has the largest mass arrives first. (b) The one that has the largest radius arrives first. (c) The hoop arrives first. (d) The disk arrives first. (e) The hoop and the disk arrive at the same time.

A solid cylinder of mass M and radius R rolls down an incline without slipping. Its moment of inertia about an axis through its center of mass is MR2/2. At any instant while in motion, its rotational kinetic energy about its center of mass is what fraction of its total kinetic energy? (a) 12 (b) 14 (c) 13 (d) 25 (e) None of these

A mouse is initially at rest on a horizontal turntable mounted on a frictionless, vertical axle. As the mouse begins to walk clockwise around the perimeter, which of the following statements must be true of the turntable? (a) It also turns clockwise. (b) It turns counterclockwise with the same angular velocity as the mouse. (c) It remains stationary. (d) It turns counterclock- wise because angular momentum is conserved. (e) It turns counterclockwise because mechanical energy is conserved.

Consider two uniform, solid spheres, a large, massive sphere and a smaller, lighter sphere. They are released from rest simultaneously from the top of a hill and roll down without slipping. Which one reaches the bottom of the hill first? (a) The large sphere reaches the bottom first. (b) The small sphere reaches the bottom first. (c) The sphere with the greatest density reaches the bottom first. (d) The spheres reach the bottom at the same time. (e) The answer depends on the values of the spheres masses and radii.

The cars in a soapbox derby have no engines; they simply coast downhill. Which of the following design criteria is best from a competitive point of view? The cars wheels should (a) have large moments of inertia, (b) be massive, (c) be hoop-like wheels rather than solid disks, (d) be large wheels rather than small wheels, or (e) have small moments of inertia.

Two ponies of equal mass are initially at diametrically opposite points on the rim of a large horizontal turntable that is turning freely on a vertical, frictionless axle through its center. The ponies simultaneously start walking toward each other across the turntable. (i) As they walk, what happens to the angular speed of the carousel? (a) It increases. (b) It decreases. (c) It stays constant. Consider the poniesturntable system in this process, and answer yes or no for the following questions. (ii) Is the mechanical energy of the system conserved? (iii) Is the momentum of the system conserved? (iv) Is the angular momentum of the system conserved?

Many of the elements in horizontal-bar exercises can be modeled by representing the gymnast by four segments consisting of arms, torso (including the head), thighs, and lower legs, as shown in Figure P8.15a. Inertial parameters for a particular gymnast are as follows: Segment Mass (kg) Length (m) rcg (m) I (kg ? m2) Arms 6.87 0.548 0.239 0.205 Torso 33.57 0.601 0.337 1.610 Thighs 14.07 0.374 0.151 0.173 Legs 7.54 0.227 0.164 Note that in Figure P8.15a rcg is the distance to the center of gravity measured from the joint closest to the bar and the masses for the arms, thighs, and legs include both appendages. I is the moment of inertia of each segment about its center of gravity. Determine the distance from the bar to the center of gravity of the gymnast for the two positions shown in Figures P8.15b and P8.15c.

Using the data given in Problem 15 and the coordinate system shown in Figure P8.16b, calculate the position of the center of gravity of the gymnast shown in Figure P8.16a. Pay close attention to the definition of rcg in the table.

A person bending forward to lift a load with his back (Fig. P8.17a) rather than with his knees can be injured by large forces exerted on the muscles and vertebrae. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, and to understand why back problems are common among humans, consider the model shown in Figure P8.17b of a person bending forward to lift a 200-N object. The spine and upper body are represented as a uniform horizontal rod of weight 350 N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point twothirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is 12.0. Find (a) the tension in the back muscle and (b) the compressional force in the spine.

?When a person stands on tiptoe (a strenuous position), the position of the foot is as shown in Figure P8.18a. The total gravitational force on the body, \(\overrightarrow{\mathbf{F}}_{g}\), is supported by the force \(\overrightarrow{\mathbf{n}}\) exerted by the floor on the toes of one foot. A mechanical model of the situation is shown in Figure P8.18b, where \(\overrightarrow{\mathbf{T}}\) is the force exerted by the Achilles tendon on the foot and \(\overrightarrow{\mathbf{R}}\) is the force exerted by the tibia on the foot. Find the values of T, R, and \(\theta\) when \(F_g=n=700\mathrm{\ N}\).

A 500-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 100-N rod as indicated in Figure P8.19. The left end of the rod is supported by a hinge, and the right end is supported by a thin cable making a 30.0 angle with the vertical. (a) Find the tension T in the cable. (b) Find the horizontal and vertical components of force exerted on the left end of the rod by the hinge.

A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs 200 N and is 3.00 m long. What is the tension in each rope when the 700-N worker stands 1.00 m from one end?

A uniform plank of length 2.00 m and mass 30.0 kg is supported by three ropes, as indicated by the blue vectors in Figure P8.21. Find the tension in each rope when a 700-N person is d 5 0.500 m from the left end.

A hungry bear weighing 700 N walks out on a beam in an attempt to retrieve a basket of goodies hanging at the end of the beam (Fig. P8.22). The beam is uniform, weighs 200 N, and is 6.00 m long, and it is supported by a wire at an angle of u 5 60.0. The basket weighs 80.0 N. (a) Draw a force diagram for the beam. (b) When the bear is at x 5 1.00 m, find the tension in the wire supporting the beam and the components of the force exerted by the wall on the left end of the beam. (c) If the wire can withstand a maximum tension of 900 N, what is the maximum distance the bear can walk before the wire breaks?

Figure P8.23 shows a uniform beam of mass m pivoted at its lower end, with a horizontal spring attached between its top end and a vertical wall. The beam makes an angle u with the horizontal. Find expressions for (a) the distance d the spring is stretched from equilibrium and (b) the components of the force exerted by the pivot on the beam.

A strut of length L 5 3.00 m and mass m 5 16.0 kg is held by a cable at an angle of u 5 30.0 with respect to the horizontal as shown in Figure P8.24. (a) Sketch a force diagram, indicating all the forces and their placement on the strut. (b) Why is the hinge a good place to use for calculating torques? (c) Write the condition for rotational equilibrium symbolically, calculating the torques around the hinge. (d) Use the torque equation to calculate the tension in the cable. (e) Write the x- and y-components of Newtons second law for equilibrium. (f) Use the force equation to find the x- and y-components of the force on the hinge. (g) Assuming the strut position is to remain the same, would it be advantageous to attach the cable higher up on the wall? Explain the benefit in terms of the force on the hinge and cable tension.

A refrigerator of width w and height h rests on a rough incline as in Figure P8.25 (page 268). Find an expression for the maximum value u can have before the refrigerator tips over. Note, the contact point between the refrigerator and incline shifts as u increases and treat the refrigerator as a uniform box. u h w Figure P8.25

A uniform beam of length L and mass m shown in Figure P8.26 is inclined at an angle u to the horizontal. Its upper end is connected to a wall by a rope, and its lower end rests on a rough horizontal surface. The coefficient of static friction between the beam and surface is ms. Assume the angle is such that the static friction force is at its maximum value. (a) Draw a force diagram for the beam. (b) Using the condition of rotational equilibrium, find an expression for the tension T in the rope in terms of m, g, and u. (c) Using Newtons second law for equilibrium, find a second expression for T in terms of ms, m, and g. (d) Using the foregoing results, obtain a relationship involving only ms and the angle u. (e) What happens if the angle gets smaller? Is this equation valid for all values of u? Explain.

The chewing muscle, the masseter, is one of the strongest in the human body. It is attached to the mandible (lower jawbone) as shown in Figure P8.27a. The jawbone is pivoted about a socket just in front of the auditory canal. The forces acting on the jawbone are equivalent to those acting on the curved bar in Figure P8.27b. F S C is the force exerted by the food being chewed against the jawbone, T S is the force of tension in the masseter, and R S is the force exerted by the socket on the mandible. Find T S and R S for a person who bites down on a piece of steak with a force of 50.0 N.

A 1 200-N uniform boom at f 5 65 to the horizontal is supported by a cable at an angle u 5 25.0 to the horizontal as shown in Figure P8.28. The boom is pivoted at the bottom, and an object of weight w 5 2 000 N hangs from its top. Find (a) the tension in the support cable and (b) the components of the reaction force exerted by the pivot on the boom.

The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Fig. P8.29a). The forces on the lower leg when the leg is extended are modeled as in Figure P8.29b, where T S is the force of tension in the tendon, wS is the force of gravity acting on the lower leg, and F S is the force of gravity acting on the foot. Find T S when the tendon is at an angle of 25.0 with the tibia, assuming that w 5 30.0 N, F 5 12.5 N, and the leg is extended at an angle u of 40.0 with the vertical. Assume that the center of gravity of the lower leg is at its center and that the tendon attaches to the lower leg at a point one-fifth of the way down the leg.

One end of a uniform 4.0-m-long rod of weight w is supported by a cable at an angle of u 5 37 with the rod. The other end rests against a wall, where it is held by friction. (See Fig. P8.30.) The coefficient of static friction between the wall and the rod is ms 5 0.50. Determine the minimum distance x from point A at which an additional weight w (the same as the weight of the rod) can be hung without causing the rod to slip at point A.

Four objects are held in position at the corners of a rectangle by light rods as shown in Figure P8.31. Find the moment of inertia of the system about (a) the x-axis, (b) the y-axis, and (c) an axis through O and perpendicular to the page.

If the system shown in Figure P8.31 is set in rotation about each of the axes mentioned in Problem 31, find the torque that will produce an angular acceleration of 1.50 rad/s2 in each case.

A large grinding wheel in the shape of a solid cylinder of radius 0.330 m is free to rotate on a frictionless, vertical axle. A constant tangential force of 250 N applied to its edge causes the wheel to have an angular acceleration of 0.940 rad/s2. (a) What is the moment of inertia of the wheel? (b) What is the mass of the wheel? (c) If the wheel starts from rest, what is its angular velocity after 5.00 s have elapsed, assuming the force is acting during that time?

An oversized yo-yo is made from two identical solid disks each of mass M 5 2.00 kg and radius R 5 10.0 cm. The two disks are joined by a solid cylinder of radius r 5 4.00 cm and mass m 5 1.00 kg as in Figure P8.34. Take the center of the cylinder as the axis of the system, with positive torques directed to the left along this axis. All torques and angular variables are to be calculated around this axis. Light string is wrapped around the cylinder, and the system is then allowed to drop from rest. (a) What is the moment of inertia of the system? Give a symbolic answer. (b) What torque does gravity exert on the system with respect to the given axis? (c) Take downward as the negative coordinate direction. As depicted in Figure P8.34, is the torque exerted by the tension positive or negative? Is the angular acceleration positive or negative? What about the translational acceleration? (d) Write an equation for the angular acceleration a in terms of the translational acceleration a and radius r. (Watch the sign!) (e) Write Newtons second law for the system in terms of m, M, a, T, and g. (f) Write Newtons second law for rotation in terms of I, a, T, and r. (g) Eliminate a from the rotational second law with the expression found in part (d) and find a symbolic expression for the acceleration a in terms of m, M, g, r and R. (h) What is the numeric value for the systems acceleration? (i) What is the tension in the string? (j) How long does it take the system to drop 1.00 m from rest?

A rope of negligible mass is wrapped around a 225-kg solid cylinder of radius 0.400 m. The cylinder is suspended several meters off the ground with its axis oriented horizontally, and turns on that axis without friction. (a) If a 75.0-kg man takes hold of the free end of the rope and falls under the force of gravity, what is his acceleration? (b) What is the angular acceleration of the cylinder? (c) If the mass of the rope were not neglected, what would happen to the angular acceleration of the cylinder as the man falls?

A potters wheel having a radius of 0.50 m and a moment of inertia of 12 kg ? m2 is rotating freely at 50 rev/min. The potter can stop the wheel in 6.0 s by pressing a wet rag against the rim and exerting a radially inward force of 70 N. Find the effective coefficient of kinetic friction between the wheel and the wet rag.

A model airplane with mass 0.750 kg is tethered by a wire so that it flies in a circle 30.0 m in radius. The airplane engine provides a net thrust of 0.800 N perpendicular to the tethering wire. (a) Find the torque the net thrust produces about the center of the circle. (b) Find the angular acceleration of the airplane when it is in level flight. (c) Find the linear acceleration of the airplane tangent to its flight path.

A bicycle wheel has a diameter of 64.0 cm and a mass of 1.80 kg. Assume that the wheel is a hoop with all the mass concentrated on the outside radius. The bicycle is placed on a stationary stand, and a resistive force of 120 N is applied tangent to the rim of the tire. (a) What force must be applied by a chain passing over a 9.00-cm-diameter sprocket in order to give the wheel an acceleration of 4.50 rad/s2? (b) What force is required if you shift to a 5.60-cm-diameter sprocket?

A 150-kg merry-go-round in the shape of a uniform, solid, horizontal disk of radius 1.50 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force must be exerted on the rope to bring the merry-go-round from rest to an angular speed of 0.500 rev/s in 2.00 s?

An Atwoods machine consists of blocks of masses m1 5 10.0 kg and m2 5 20.0 kg attached by a cord running over a pulley as in Figure P8.40. The pulley is a solid cylinder with mass M 5 8.00 kg and radius r 5 0.200 m. The block of mass m2 is allowed to drop, and the cord turns the pulley without slipping. (a) Why must the tension T2 be greater than the tension T1? (b) What is the acceleration of the system, assuming the pulley axis is frictionless? (c) Find the tensions T1 and T2.

An airliner lands with a speed of 50.0 m/s. Each wheel of the plane has a radius of 1.25 m and a moment of inertia of 110 kg ? m2. At touchdown, the wheels begin to spin under the action of friction. Each wheel supports a weight of 1.40 3 104 N, and the wheels attain their angular speed in 0.480 s while rolling without slipping. What is the coefficient of kinetic friction between the wheels and the runway? Assume that the speed of the plane is constant.

A car is designed to get its energy from a rotating flywheel with a radius of 2.00 m and a mass of 500 kg. Before a trip, the flywheel is attached to an electric motor, which brings the flywheels rotational speed up to 5 000 rev/min. (a) Find the kinetic energy stored in the flywheel. (b) If the flywheel is to supply energy to the car as a 10.0-hp motor would, find the length of time the car could run before the flywheel would have to be brought back up to speed.

A horizontal 800-N merry-go-round of radius 1.50 m is started from rest by a constant horizontal force of 50.0N applied tangentially to the merry-go-round. Find the kinetic energy of the merry-go-round after 3.00 s. (Assume it is a solid cylinder.)

Four objectsa hoop, a solid cylinder, a solid sphere, and a thin, spherical shelleach have a mass of 4.80 kg and a radius of 0.230 m. (a) Find the moment of inertia for each object as it rotates about the axes shown in Table 8.1. (b) Suppose each object is rolled down a ramp. Rank the translational speed of each object from highest to lowest. (c) Rank the objects rotational kinetic energies from highest to lowest as the objects roll down the ramp.

A light rod of length , 5 1.00 m rotates about an axis perpendicular to its length and passing through its center as in Figure P8.45. Two particles of masses m1 5 4.00 kg and m2 5 3.00 kg are connected to the ends of the rod. (a) Neglecting the mass of the rod, what is the systems kinetic energy when its angular speed is 2.50 rad/s? (b) Repeat the problem, assuming the mass of the rod is taken to be 2.00 kg.

A 240-N sphere 0.20 m in radius rolls without slipping 6.0 m down a ramp that is inclined at 37 with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest?

A solid, uniform disk of radius 0.250 m and mass 55.0 kg rolls down a ramp of length 4.50 m that makes an angle of 15.0 with the horizontal. The disk starts from rest from the top of the ramp. Find (a) the speed of the disks center of mass when it reaches the bottom of the ramp and (b) the angular speed of the disk at the bottom of the ramp.

A solid uniform sphere of mass m and radius R rolls without slipping down an incline of height h. (a) What forms of mechanical energy are associated with the sphere at any point along the incline when its angular speed is v? Answer in words and symbolically in terms of the quantities m, g, y, I, v, and v. (b) What force acting on the sphere causes it to roll rather than slip down the incline? (c) Determine the ratio of the spheres rotational kinetic energy to its total kinetic energy at any instant.

The top in Figure P8.49 has a moment of inertia of 4.00 3 1024 kg ? m2 and is initially at rest. It is free to rotate about a stationary axis AA9. A string wrapped around a peg along the axis of the top is pulled in such a manner as to maintain a constant tension of 5.57 N in the string. If the string does not slip while wound around the peg, what is the angular speed of the top after 80.0 cm of string has been pulled off the peg? Hint: Consider the work that is done.

A constant torque of 25.0 N ? m is applied to a grindstone whose moment of inertia is 0.130 kg ? m2. Using energy principles and neglecting friction, find the angular speed after the grindstone has made 15.0 revolutions. Hint: The angular equivalent of Wnet 5 FDx 5 12 mvf 2 2 12 mvi 2 is Wnet 5 tDu 5 12 Ivf 2 2 12 Ivi 2. You should convince yourself that this last relationship is correct.

A 10.0-kg cylinder rolls without slipping on a rough surface. At an instant when its center of gravity has a speed of 10.0 m/s, determine (a) the translational kinetic energy of its center of gravity, (b) the rotational kinetic energy about its center of gravity, and (c) its total kinetic energy.

Use conservation of energy to determine the angular speed of the spool shown in Figure P8.52 after the 3.00-kg bucket has fallen 4.00 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

A giant swing at an amusement park consists of a 365-kg uniform arm 10.0 m long, with two seats of negligible mass connected at the lower end of the arm (Fig. P8.53). (a) How far from the upper end is the center of mass of the arm? (b) The gravitational potential energy of the arm is the same as if all its mass were concentrated at the center of mass. If the arm is raised through a 45.0 angle, find the gravitational potential energy, where the zero level is taken to be 10.0 m below the axis. (c) The arm drops from rest from the position described in part (b). Find the gravitational potential energy of the system when it reaches the vertical orientation. (d) Find the speed of the seats at the bottom of the swing. 10.0 m Figure P8.53

Each of the following objects has a radius of 0.180 m and a mass of 2.40 kg, and each rotates about an axis through its center (as in Table 8.1) with an angular speed of 35.0 rad/s. Find the magnitude of the angular momentum of each object. (a) a hoop (b) a solid cylinder (c) a solid sphere (d) a hollow spherical shell

(a) Calculate the angular momentum of Earth that arises from its spinning motion on its axis, treating Earth as a uniform solid sphere. (b) Calculate the angular momentum of Earth that arises from its orbital motion about the Sun, treating Earth as a point particle.

A 0.005 00-kg bullet traveling horizontally with a speed of 1.00 3 103 m/s enters an 18.0-kg door, embedding itself 10.0 cm from the side opposite the hinges as in Figure P8.56. The 1.00-m-wide door is free to swing on its hinges. (a) Before it hits the door, does the bullet have angular momentum relative the doors axis of rotation? Explain. (b) Is mechanical energy conserved in this collision? Answer without doing a calculation. (c) At what angular speed does the door swing open immediately after the collision? (The door has the same moment of inertia as a rod with axis at one end.) (d) Calculate the energy of the doorbullet system and determine whether it is less than or equal to the kinetic energy of the bullet before the collision.

A light rigid rod of length , 5 1.00 m rotates about an axis perpendicular to its length and through its center, as shown in Figure P8.45. Two particles of masses m1 5 4.00 kg and m2 5 3.00 kg are connected to the ends of the rod. What is the angular momentum of the system if the speed of each particle is 5.00 m/s? (Neglect the rods mass.)

Halleys comet moves about the Sun in an elliptical orbit, with its closest approach to the Sun being 0.59 A.U. and its greatest distance being 35 A.U. (1 A.U. is the EarthSun distance). If the comets speed at closest approach is 54 km/s, what is its speed when it is farthest from the Sun? You may neglect any change in the comets mass and assume that its angular momentum about the Sun is conserved.

A rigid, massless rod has three particles with equal masses attached to it as shown in Figure P8.59. The rod is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through the point P and is released from rest in the horizontal position at t 5 0. Assuming m and d are known, find (a) the moment of inertia of the system (rod plus particles) about the pivot, (b) the torque acting on the system at t 5 0, (c) the angular acceleration of the system at t 5 0, (d) the linear acceleration of the particle labeled 3 at t 5 0, (e) the maximum kinetic energy of the system, (f) the maximum angular speed reached by the rod, (g) the maximum angular momentum of the system, and (h) the maximum translational speed reached by the particle labeled 2.

A 60.0-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 500 kg ? m2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to Earth. (a) In what direction and with what angular speed does the turntable rotate? (b) How much work does the woman do to set herself and the turntable into motion?

A solid, horizontal cylinder of mass 10.0 kg and radius 1.00 m rotates with an angular speed of 7.00 rad/s about a fixed vertical axis through its center. A 0.250-kg piece of putty is dropped vertically onto the cylinder at a point 0.900 m from the center of rotation and sticks to the cylinder. Determine the final angular speed of the system.

A student sits on a rotating stool holding two 3.0-kg objects. When his arms are extended horizontally, the objects are 1.0 m from the axis of rotation and he rotates with an angular speed of 0.75 rad/s. The moment of inertia of the student plus stool is 3.0 kg ? m2 and is assumed to be constant. The student then pulls in the objects horizontally to 0.30 m from the rotation axis. (a) Find the new angular speed of the student. (b) Find the kinetic energy of the student before and after the objects are pulled in.

The puck in Figure P8.63 has a mass of 0.120 kg. Its original distance from the center of rotation is 40.0 cm, and it moves with a speed of 80.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. Determine the work done on the puck. Hint: Consider the change in kinetic energy of the puck.

A space station shaped like a giant wheel has a radius of 100 m and a moment of inertia of 5.00 3 108 kg ? m2. A crew of 150 lives on the rim, and the station is rotating so that the crew experiences an apparent acceleration of 1g (Fig. P8.64). When 100 people move to the center of the station for a union meeting, the angular speed changes. What apparent acceleration is experienced by the managers remaining at the rim? Assume the average mass of a crew member is 65.0 kg.

A cylinder with moment of inertia I1 rotates with angular velocity v0 about a frictionless vertical axle. A second cylinder, with moment of inertia I2, initially not rotating, drops onto the first cylinder (Fig. P8.65). Because the surfaces are rough, the two cylinders eventually reach the same angular speed v. (a) Calculate v. (b) Show that kinetic energy is lost in this situation, and calculate the ratio of the final to the initial kinetic energy.

A particle of mass 0.400 kg is attached to the 100-cm mark of a meter stick of mass 0.100 kg. The meter stick rotates on a horizontal, frictionless table with an angular speed of 4.00 rad/s. Calculate the angular momentum of the system when the stick is pivoted about an axis (a) perpendicular to the table through the 50.0-cm mark and (b) perpendicular to the table through the 0-cm mark.

A typical propeller of a turbine used to generate electricity from the wind consists of three blades as in Figure P8.67. Each blade has a length of L 5 35 m and a mass of m 5 420 kg. The propeller rotates at the rate of 25 rev/min. (a) Convert the angular speed of the propeller to units of rad/s. Find (b) the moment of inertia of the propeller about the axis of rotation and (c) the total kinetic energy of the propeller.

Figure P8.68 shows a clawhammer as it is being used to pull a nail out of a horizontal board. If a force of magnitude 150 N is exerted horizontally as shown, find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface at the point of contact with the hammer head. Assume that the force the hammer exerts on the nail is parallel to the nail and perpendicular to the position vector from the point of contact.

A 40.0-kg child stands at one end of a 70.0-kg boat that is 4.00 m long (Fig. P8.69). The boat is initially 3.00 m from the pier. The child notices a turtle on a rock beyond the far end of the boat and proceeds to walk to that end to catch the turtle. (a) Neglecting friction between the boat and water, describe the motion of the system (child plus boat). (b) Where will the child be relative to the pier when he reaches the far end of the boat? (c) Will he catch the turtle? (Assume that he can reach out 1.00 m from the end of the boat.)

An object of mass M 5 12.0 kg is attached to a cord that is wrapped around a wheel of radius r 5 10.0 cm (Fig. P8.70). The acceleration of the object down the frictionless incline is measured to be a 5 2.00 m/s2 and the incline makes an angle u 5 37.0 with the horizontal. Assuming the axle of the wheel to be frictionless, determine (a) the tension in the rope, (b) the moment of inertia of the wheel, and (c) the angular speed of the wheel 2.00 s after it begins rotating, starting from rest.

A uniform ladder of length L and weight w is leaning against a vertical wall. The coefficient of static friction between the ladder and the floor is the same as that between the ladder and the wall. If this coefficient of static friction is ms 5 0.500, determine the smallest angle the ladder can make with the floor without slipping.

Two astronauts (Fig. P8.72), each having a mass of 75.0 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.00 m/s. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum and (b) the rotational energy of the system. By pulling on the rope, the astronauts shorten the distance between them to 5.00 m. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new rotational energy of the system? (f) How much work is done by the astronauts in shortening the rope?

This is a symbolic version of problem 72. Two astronauts (Fig. P8.72), each having a mass M, are connected by a rope of length d having negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed v. (a) Calculate the magnitude of the angular momentum of the system by treating the astronauts as particles. (b) Calculate the rotational energy of the system. By pulling on the rope, the astronauts shorten the distance between them to d/2. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new rotational energy of the system? (f) How much work is done by the astronauts in shortening the rope?

Two window washers, Bob and Joe, are on a 3.00-m-long, 345-N scaffold supported by two cables attached to its ends. Bob weighs 750 N and stands 1.00 m from the left end, as shown in Figure P8.74. Two meters from the left end is the 500-N washing equipment. Joe is 0.500 m from the right end and weighs 1 000 N. Given that the scaffold is in rotational and translational equilibrium, what are the forces on each cable?

A 2.35-kg uniform bar of length , 5 1.30 m is held in a horizontal position by three vertical springs as in Figure P8.75. The two lower springs are compressed and exert upward forces on the bar of magnitude F1 5 6.80 N and F2 5 9.50 N, respectively. Find (a) the force Fs exerted by the top spring on the bar, and (b) the location x of the upper spring that will keep the bar in equilibrium.

A light rod of length 2L is free to rotate in a vertical plane about a frictionless pivot through its center. A particle of mass m1 is attached at one end of the rod, and a mass m2 is at the opposite end, where m1 . m2. The system is released from rest in the vertical position shown in Figure P8.76a (page 274), and at some later time the system is rotating in the position shown in Figure P8.76b. Take the reference point of the gravitational potential energy to be at the pivot. (a) Find an expression for the systems total mechanical energy in the vertical position. (b) Find an expression for the total mechanical energy in the rotated position shown in Figure P8.76b. (c) Using the fact that the mechanical energy of the system is conserved, how would you determine the angular speed v of the system in the rotated position? (d) Find the magnitude of the torque on the system in the vertical position and in the rotated position. Is the torque constant? Explain what these results imply regarding the angular momentum of the system. (e) Find an expression for the magnitude of the angular acceleration of the system in the rotated position. Does your result make sense when the rod is horizontal? When it is vertical? Explain. Figure P8.76

A light rope passes over a light, frictionless pulley. One end is fastened to a bunch of bananas of mass M, and a monkey of mass M clings to the other end (Fig. P8.77). The monkey climbs the rope in an attempt to reach the bananas. (a) Treating the system as consisting of the monkey, bananas, rope, and pulley, find the net torque of the system about the pulley axis. (b) Using the result of part (a), determine the total angular momentum about the pulley axis and describe the motion of the system. (c) Will the monkey reach the bananas before they get stuck in the pulley?

An electric motor turns a flywheel through a drive belt that joins a pulley on the motor and a pulley that is rigidly attached to the flywheel, as shown in Figure P8.78. The flywheel is a uniform disk with a mass of 80.0 kg and a radius of R 5 0.625 m. It turns on a frictionless axle. Its pulley has much smaller mass and a radius of r 5 0.230 m. The tension Tu in the upper (taut) segment of the belt is 135 N and the flywheel has a clockwise angular acceleration of 1.67 rad/s2. Find the tension in the lower (slack) segment of the belt.

In exercise physiology studies, it is sometimes important to determine the location of a persons center of gravity. This can be done with the arrangement shown in Figure P8.79. A light plank rests on two scales that read Fg1 5 380 N and Fg2 5 320 N. The scales are separated by a distance of 2.00 m. How far from the womans feet is her center of gravity?

A uniform thin rod of length L and mass M is free to rotate on a frictionless pin passing through one end (Fig. P8.80). The rod is released from rest in the horizontal position. (a) What is the speed of its center of gravity when the rod reaches its lowest position? (b) What is the tangential speed of the lowest point on the rod when it is in the vertical position?

A uniform solid cylinder of mass M and radius R rotates on a frictionless horizontal axle (Fig. P8.81). Two objects with equal masses m hang from light cords wrapped around the cylinder. If the system is released from rest, find (a) the tension in each cord and (b) the acceleration of each object after the objects have descended a distance h.

A painter climbs a ladder leaning against a smooth wall. At a certain height, the ladder is on the verge of slipping. (a) Explain why the force exerted by the vertical wall on the ladder is horizontal. (b) If the ladder of length L leans at an angle u with the horizontal, what is the lever arm for this horizontal force with the axis of rotation taken at the base of the ladder? (c) If the ladder is uniform, what is the lever arm for the force of gravity acting on the ladder? (d) Let the mass of the painter be 80 kg, L 5 4.0 m, the ladders mass be 30 kg, u 5 53, and the coefficient of friction between ground and ladder be 0.45. Find the maximum distance the painter can climb up the ladder.

A war-wolf, or trebuchet, is a device used during the Middle Ages to throw rocks at castles and now sometimes used to fling pumpkins and pianos. A simple trebuchet is shown in Figure P8.83. Model it as a stiff rod of negligible mass 3.00 m long and joining particles of mass m1 5 0.120 kg and m2 5 60.0 kg at its ends. It can turn on a frictionless horizontal axle perpendicular to the rod and 14.0 cm from the particle of larger mass. The rod is released from rest in a horizontal orientation. Find the maximum speed that the object of smaller mass attains. 3.00 m m1 m2 Figure P8.83

A string is wrapped around a uniform cylinder of mass M and radius R. The cylinder is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P8.84). Show that (a) the tension in the string is one-third the weight of the cylinder, (b) the magnitude of the acceleration of the center of gravity is 2g/3, and (c) the speed of the center of gravity is (4gh/3)1/2 after the cylinder has descended through distance h. Verify your answer to part (c) with the energy approach.

The Iron Cross When a gymnast weighing 750 N executes the iron cross as in Figure P8.85a, the primary muscles involved in supporting this position are the latissimus dorsi (lats) and the pectoralis major (pecs). The rings exert an upward force on the arms and support the weight of the gymnast. The force exerted by the shoulder joint on the arm is labeled F S s while the two muscles exert a total force F S m on the arm. Estimate the magnitude of the force F S m. Note that one ring supports half the weight of the gymnast, which is 375 N as indicated in Figure P8.85b. Assume that the force F S m acts at an angle of 45 below the horizontal at a distance of 4.0 cm from the shoulder joint. In your estimate, take the distance from the shoulder joint to the hand to be L 5 70 cm and ignore the weight of the arm.

In an emergency situation, a person with a broken forearm ties a strap from his hand to clip on his shoulder as in Figure P8.86. His 1.60- kg forearm remains in a horizontal position and the strap makes an angle of u 5 50.0 with the horizontal. Assume the forearm is uniform, has a length of , 5 0.320 m, assume the biceps muscle is relaxed, and ignore the mass and length of the hand. Find (a) the tension in the strap and (b) the components of the reaction force exerted by the humerus on the forearm.

An object of mass m1 5 4.00 kg is connected by a light cord to an object of mass m2 5 3.00 kg on a frictionless surface (Fig. P8.87). The pulley rotates about a frictionless axle and has a moment of inertia of 0.500 kg ? m2 and a radius of 0.300 m. Assuming that the cord does not slip on the pulley, find (a) the acceleration of the two masses and (b) the tensions T1 and T2.

A 10.0-kg monkey climbs a uniform ladder with weight w 5 1.20 3 102 N and length L 5 3.00 m as shown in Figure P8.88 (page 276). The ladder rests against the wall at an angle of u 5 60.0. The upper and lower ends of the ladder rest on frictionless surfaces, with the lower end fastened to the wall by a horizontal rope that is frayed and that can support a maximum tension of only 80.0 N. (a) Draw a force diagram for the ladder. (b) Find the normal force exerted by the bottom of the ladder. (c) Find the tension in the rope when the monkey is twothirds of the way up the ladder. (d) Find the maximum distance d that the monkey can climb up the ladder before the rope breaks. (e) If the horizontal surface were rough and the rope were removed, how would your analysis of the problem be changed and what other information would you need to answer parts (c) and (d)?

A 3.2-kg sphere is suspended by a cord that passes over a 1.8-kg pulley of radius 3.8 cm. The cord is attached to a spring whose force constant is k 5 86 N/m as in Figure P8.89. Assume the pulley is a solid disk. (a) If the sphere is released from rest with the spring unstretched, what distance does the sphere fall through before stopping? (b) Find the speed of the sphere after it has fallen 25 cm.