A pair of forces with equal magnitudes, opposite

The wheel of a car has a radius of 0.350 m. The engine of the car applies a torque of 295 N ? m to this wheel, which does not slip against the road surface. Since the wheel does not slip, the road must be applying a force of static friction to the wheel that produces a countertorque. Moreover, the car has a constant velocity, so this countertorque balances the applied torque. What is the magnitude of the static frictional force?

The steering wheel of a car has a radius of 0.19 m, and the steering wheel of a truck has a radius of 0.25 m. The same force is applied in the same direction to each steering wheel. What is the ratio of the torque produced by this force in the truck to the torque produced in the car?

You are installing a new spark plug in your car, and the manual speci- fi es that it be tightened to a torque that has a magnitude of 45 N ? m. Using the data in the drawing, determine the magnitude F of the force that you must exert on the wrench.

Two children hang by their hands from the same tree branch. The branch is straight, and grows out from the tree trunk at an angle of 27.08 above the horizontal. One child, with a mass of 44.0 kg, is hanging 1.30 m along the branch from the tree trunk. The other child, with a mass of 35.0 kg, is hanging 2.10 m from the tree trunk. What is the magnitude of the net torque exerted on the branch by the children? Assume that the axis is located where the branch joins the tree trunk and is perpendicular to the plane formed by the branch and the trunk.

The drawing shows a jet engine suspended beneath the wing of an airplane. The weight W B of the engine is 10 200 N and acts as shown in the drawing. In fl ight the engine produces a thrust T B of 62 300 N that is parallel to the ground. The rotational axis in the drawing is perpendicular to the plane of the paper. With respect to this axis, fi nd the magnitude of the torque due to (a) the weight and (b) the thrust.

A square, 0.40 m on a side, is mounted so that it can rotate about an axis that passes through the center of the square. The axis is perpendicular to the plane of the square. A force of 15 N lies in this plane and is applied to the square. What is the magnitude of the maximum torque that such a force could produce?

A pair of forces with equal magnitudes, opposite directions, and diff erent lines of action is called a couple. When a couple acts on a rigid object, the couple produces a torque that does not depend on the location of the axis. The drawing shows a couple acting on a tire wrench, each force being perpendicular to the wrench. Determine an expression for the torque produced by the couple when the axis is perpendicular to the tire and passes through (a) point A, (b) point B, and (c) point C. Express your answers in terms of the magnitude F of the force and the length L of the wrench.

One end of a meter stick is pinned to a table, so the stick can rotate freely in a plane parallel to the tabletop. Two forces, both parallel to the tabletop, are applied to the stick in such a way that the net torque is zero. The fi rst force has a magnitude of 2.00 N and is applied perpendicular to the length of the stick at the free end. The second force has a magnitude of 6.00 N and acts at a 30.08 angle with respect to the length of the stick. Where along the stick is the 6.00-N force applied? Express this distance with respect to the end of the stick that is pinned.

A rod is lying on the top of a table. One end of the rod is hinged to the table so that the rod can rotate freely on the tabletop. Two forces, both parallel to the tabletop, act on the rod at the same place. One force is directed perpendicular to the rod and has a magnitude of 38.0 N. The second force has a magnitude of 55.0 N and is directed at an angle u with respect to the rod. If the sum of the torques due to the two forces is zero, what must be the angle u?

A rotational axis is directed perpendicular to the plane of a square and is located as shown in the drawing. Two forces, F1 B and F2 B , are applied to diagonally opposite corners, and act along the sides of the square, fi rst as shown in part a and then as shown in part b of the drawing. In each case the net torque produced by the forces is zero. The square is one meter on a side, and the magnitude of F2 B is three times that of F1 B . Find the distances a and b that locate the axis.

A person is standing on a level fl oor. His head, upper torso, arms, and hands together weigh 438 N and have a center of gravity that is 1.28 m above the fl oor. His upper legs weigh 144 N and have a center of gravity that is 0.760 m above the fl oor. Finally, his lower legs and feet together weigh 87 N and have a center of gravity that is 0.250 m above the fl oor. Relative to the fl oor, fi nd the location of the center of gravity for his entire body.

The drawing shows a per- son (weight, W 5 584 N) doing push-ups. Find the normal force exerted by the fl oor on each hand and each foot, assuming that the person holds this position

A hiker, who weighs 985 N, is strolling through the woods and crosses a small horizontal bridge. The bridge is uniform, weighs 3610 N, and rests on two concrete supports, one at each end. He stops one-fi fth of the way along the bridge. What is the magnitude of the force that a concrete support exerts on the bridge (a) at the near end and (b) at the far end?

Conceptual Example 7 provides useful background for this problem. Workers have loaded a delivery truck in such a way that its center of gravity is only slightly forward of the rear axle, as shown in the drawing. The mass of the truck and its contents is 7460 kg. Find the magnitudes of the forces exerted by the ground on (a) the front wheels and (b) the rear wheels of the truck

A person exerts a horizontal force of 190 N in the test apparatus shown in the drawing. Find the horizontal force M B (magnitude and direction) that his fl exor muscle exerts on his forearm

The drawing shows a rectangular piece of wood. The forces applied to corners B and D have the same magnitude of 12 N and are directed parallel to the long and short sides of the rectangle. The long side of the rectangle is twice as long as the short side. An axis of rotation is shown perpendicular to the plane of the rectangle at its center. A third force (not shown in the drawing) is applied to corner A, directed along the short side of the rectangle (either toward B or away from B), such that the piece of wood is at equilibrium. Find the magnitude and direction of the force applied to corner A.

Review Multiple-Concept Example 8 before beginning this problem. A sport utility vehicle (SUV) and a sports car travel around the same horizontal curve. The SUV has a static stability factor of 0.80 and can negotiate the curve at a maximum speed of 18 m/s without rolling over. The sports car has a static stability factor of 1.4. At what maximum speed can the sports car negotiate the curve without rolling over?

The wheels, axle, and handles of a wheelbarrow weigh 60.0 N. The load chamber and its contents weigh 525 N. The drawing shows these two forces in two diff erent wheelbarrow designs. To support the wheelbarrow in equilibrium, the mans hands apply a force F B to the handles that is directed vertically upward. Consider a rotational axis at the point where the tire contacts the ground, directed perpendicular to the plane of the paper. Find the magnitude of the mans force for both designs.

Review Conceptual Example 7 as background material for this problem. A jet transport has a weight of 1.00 3 106 N and is at rest on the runway. The two rear wheels are 15.0 m behind the front wheel, and the planes center of gravity is 12.6 m behind the front wheel. Determine the normal force exerted by the ground on (a) the front wheel and on (b) each of the two rear wheels.

See Example 4 for data pertinent to this problem. What is the minimum value for the coeffi cient of static friction between the ladder and the ground, so that the ladder (with the fi reman on it) does not slip?

The drawing shows a uniform horizontal beam attached to a vertical wall by a frictionless hinge and supported from below at an angle u 5 398 by a brace that is attached to a pin. The beam has a weight of 340 N. Three additional forces keep the beam in equilibrium. The brace applies a force P B to the right end of the beam that is directed upward at the angle u with respect to the horizontal. The hinge applies a force to the left end of the beam that has a horizontal component H B and a vertical component V B . Find the magnitudes of these three forces.

A man holds a 178-N ball in his hand, with the forearm horizontal (see the drawing). He can support the ball in this position because of the fl exor muscle force M B, which is applied perpendicular to the forearm. The forearm weighs 22.0 N and has a center of gravity as indicated. Find (a) the magnitude of M B and (b) the magnitude and direction of the force applied by the upper arm bone to the forearm at the elbow joint.

A uniform board is leaning against a smooth vertical wall. The board is at an angle u above the horizontal ground. The coeffi cient of static friction between the ground and the lower end of the board is 0.650. Find the smallest value for the angle u, such that the lower end of the board does not slide along the ground.

The drawing shows a bicycle wheel resting against a small step whose height is h 5 0.120 m. The weight and radius of the wheel are W 5 25.0 N and r 5 0.340 m, respectively. A horizontal force F B is applied to the axle of the wheel. As the magnitude of F B increases, there comes a time when the wheel just begins to rise up and loses contact with the ground. What is the magnitude of the force when this happens?

A 1220-N uniform beam is attached to a vertical wall at one end and is supported by a cable at the other end. A 1960-N crate hangs from the far end of the beam. Using the data shown in the drawing, fi nd (a) the magnitude of the tension in the wire and (b) the magnitudes of the horizontal and vertical components of the force that the wall exerts on the left end of the beam

A person is sitting with one leg outstretched and stationary, so that it makes an angle of 30.08 with the horizontal, as the drawing indicates. The weight of the leg below the knee is 44.5 N, with the center of gravity located below the knee joint. The leg is being held in this position because of the force M B applied by the quadriceps muscle, which is attached 0.100m below the knee joint (see the drawing). Obtain the magnitude of M B.

A wrecking ball (weight 5 4800 N) is supported by a boom, which may be assumed to be uniform and has a weight of 3600 N. As the drawing shows, a support cable runs from the top of the boom to the tractor. The angle between the support cable and the horizontal is 328, and the angle between the boom and the horizontal is 488. Find (a) the tension in the support cable and (b) the magnitude of the force exerted on the lower end of the boom by the hinge at point P.

A man drags a 72-kg crate across the fl oor at a constant velocity by pulling on a strap attached to the bottom of the crate. The crate is tilted 25 above the horizontal, and the strap is inclined 61 above the horizontal. The center of gravity of the crate coincides with its geometrical center, as indicated in the drawing. Find the magnitude of the tension in the strap.

Two vertical walls are separated by a distance of 1.5 m, as the drawing shows. Wall 1 is smooth, while wall 2 is not smooth. A uniform board is propped between them. The coeffi cient of static friction between the board and wall 2 is 0.98. What is the length of the longest board that can be propped between the walls?

The drawing shows an A-shaped stepladder. Both sides of the ladder are equal in length. This ladder is standing on a frictionless horizontal surface, and only the crossbar (which has a negligible mass) of the A keeps the ladder from collapsing. The ladder is uniform and has a mass of 20.0 kg. Determine the tension in the crossbar of the ladder.

Consult Multiple-Concept Example 10 to review an approach to problems such as this. A CD has a mass of 17 g and a radius of 6.0 cm. When inserted into a player, the CD starts from rest and accelerates to an angular velocity of 21 rad/s in 0.80 s. Assuming the CD is a uniform solid disk, determine the net torque acting on it.

A clay vase on a potters wheel experiences an angular acceleration of 8.00 rad/s2 due to the application of a 10.0-N ? m net torque. Find the total moment of inertia of the vase and potters wheel.

A solid circular disk has a mass of 1.2 kg and a radius of 0.16 m. Each of three identical thin rods has a mass of 0.15 kg. The rods are attached perpendicularly to the plane of the disk at its outer edge to form a three-legged stool (see the drawing). Find the moment of inertia of the stool with respect to an axis that is perpendicular to the plane of the disk at its center. (Hint: When considering the moment of inertia of each rod, note that all of the mass of each rod is located at the same perpendicular distance from the axis.)

A ceiling fan is turned on and a net torque of 1.8 N ? m is applied to the blades. The blades have a total moment of inertia of 0.22 kg ? m2 . What is the angular acceleration of the blades?

Multiple-Concept Example 10 provides one model for solving this type of problem. Two wheels have the same mass and radius of 4.0 kg and 0.35 m, respectively. One has the shape of a hoop and the other the shape of a solid disk. The wheels start from rest and have a constant angular acceleration with respect to a rotational axis that is perpendicular to the plane of the wheel at its center. Each turns through an angle of 13 rad in 8.0 s. Find the net external torque that acts on each wheel.

A 9.75-m ladder with a mass of 23.2 kg lies fl at on the ground. A painter grabs the top end of the ladder and pulls straight upward with a force of 245 N. At the instant the top of the ladder leaves the ground, the ladder experiences an angular acceleration of 1.80 rad/s2 about an axis passing through the bottom end of the ladder. The ladders center of gravity lies halfway between the top and bottom ends. (a) What is the net torque acting on the ladder? (b) What is the ladders moment of inertia?

Multiple-Concept Example 10 off ers useful background for problems like this. A cylinder is rotating about an axis that passes through the center of each circular end piece. The cylinder has a radius of 0.0830 m, an angular speed of 76.0 rad/s, and a moment of inertia of 0.615 kg ? m2 . A brake shoe presses against the surface of the cylinder and applies a tangential frictional force to it. The frictional force reduces the angular speed of the cylinder by a factor of two during a time of 6.40 s. (a) Find the magnitude of the angular deceleration of the cylinder. (b) Find the magnitude of the force of friction applied by the brake shoe.

A long, thin rod is cut into two pieces, one being twice as long as the other. To the midpoint of piece A (the longer piece), piece B is attached perpendicularly, in order to form the inverted T shown in the drawing. The application of a net external torque causes this object to rotate about axis 1 with an angular acceleration of 4.6 rad/s2 . When the same net external torque is used to cause the object to rotate about axis 2, what is the angular acceleration?

A particle is located at each corner of an imaginary cube. Each edge of the cube is 0.25 m long, and each particle has a mass of 0.12 kg. What is the moment of inertia of these particles with respect to an axis that lies along one edge of the cube?

Multiple-Concept Example 10 reviews the approach and some of the concepts that are pertinent to this problem. The drawing shows a model for the motion of the human forearm in throwing a dart. Because of the force M B applied by the triceps muscle, the forearm can rotate about an axis at the elbow joint. Assume that the forearm has the dimensions shown in the drawing and a moment of inertia of 0.065 kg ? m2 (including the eff ect of the dart) relative to the axis at the elbow. Assume also that the force M B acts perpendicular to the forearm. Ignoring the eff ect of gravity and any frictional forces, determine the magnitude of the force M B needed to give the dart a tangential speed of 5.0 m/s in 0.10 s, starting from rest.

Two thin rectangular sheets (0.20 m 3 0.40 m) are identical. In the fi rst sheet the axis of rotation lies along the 0.20-m side, and in the second it lies along the 0.40-m side. The same torque is applied to each sheet. The fi rst sheet, starting from rest, reaches its fi nal angular velocity in 8.0 s. How long does it take for the second sheet, starting from rest, to reach the same angular velocity?

A 15.0-m length of hose is wound around a reel, which is initially at rest. The moment of inertia of the reel is 0.44 kg ? m2 , and its radius is 0.160 m. When the reel is turning, friction at the axle exerts a torque of magnitude 3.40 N ? m on the reel. If the hose is pulled so that the tension in it remains a constant 25.0 N, how long does it take to completely unwind the hose from the reel? Neglect the mass and thickness of the hose on the reel, and assume that the hose unwinds without slipping.

The drawing shows two identical systems of objects; each consists of the same three small balls connected by massless rods. In both systems the axis is perpendicular to the page, but it is located at a diff erent place, as shown. The same force of magnitude F is applied to the same ball in each system (see the drawing). The masses of the balls are m1 5 9.00 kg, m2 5 6.00 kg, and m3 5 7.00 kg. The magnitude of the force is F 5 424 N. (a) For each of the two systems, determine the moment of inertia about the given axis of rotation. (b) Calculate the torque (magnitude and direction) acting on each system. (c) Both systems start from rest, and the direction of the force moves with the system and always points along the 4.00-m rod. What is the angular velocity of each system after 5.00 s?

The drawing shows the top view of two doors. The doors are uniform and identical. Door A rotates about an axis through its left edge, and door B rotates about an axis through its center. The same force F B is applied perpendicular to each door at its right edge, and the force remains perpendicular as the door turns. No other force aff ects the rotation of either door. Starting from rest, door A rotates through a certain angle in 3.00 s. How long does it take door B (also starting from rest) to rotate through the same angle?

A stationary bicycle is raised off the ground, and its front wheel (m 5 1.3 kg) is rotating at an angular velocity of 13.1 rad/s (see the drawing). The front brake is then applied for 3.0 s, and the wheel slows down to 3.7 rad/s. Assume that all the mass of the wheel is concentrated in the rim, the radius of which is 0.33 m. The coeffi cient of kinetic friction between each brake pad and the rim is mk 5 0.85. What is the magnitude of the normal force that each brake pad applies to the rim?

The parallel axis theorem provides a useful way to calculate the moment of inertia I about an arbitrary axis. The theorem states that I 5 Icm 1 Mh2 , where Icm is the moment of inertia of the object relative to an axis that passes through the center of mass and is parallel to the axis of interest, M is the total mass of the object, and h is the perpendicular distance between the two axes. Use this theorem and information to determine an expression for the moment of inertia of a solid cylinder of radius R relative to an axis that lies on the surface of the cylinder and is perpendicular to the circular ends.

The crane shown in the drawing is lifting a 180-kg crate upward with an acceleration of 1.2 m/s2 . The cable from the crate passes over a solid cylindrical pulley at the top of the boom. The pulley has a mass of 130 kg. The cable is then wound onto a hollow cylindrical drum that is mounted on the deck of the crane. The mass of the drum is 150 kg, and its radius is 0.76 m. The engine applies a counterclockwise torque to the drum in order to wind up the cable. What is the magnitude of this torque? Ignore the mass of the cable.

Calculate the kinetic energy that the earth has because of (a) its rotation about its own axis and (b) its motion around the sun. Assume that the earth is a uniform sphere and that its path around the sun is circular. For comparison, the total energy used in the United States in one year is about 1.1 3 1020 J.

Three objects lie in the x, y plane. Each rotates about the z axis with an angular speed of 6.00 rad/s. The mass m of each object and its perpendicular distance r from the z axis are as follows: (1) m1 5 6.00 kg and r1 5 2.00 m, (2) m2 5 4.00 kg and r2 5 1.50 m, (3) m3 5 3.00 kg and r3 5 3.00 m. (a) Find the tangential speed of each object. (b) Determine the total kinetic energy of this system using the expression KE 5 1 2m1v 2 1 1 1 2m2v 2 2 1 1 2m3v 2 3 . (c) Obtain the moment of inertia of the system. (d) Find the rotational kinetic energy of the system using the relation KER 5 1 2 Iv2 to verify that the answer is the same as the answer to (b).

Two thin rods of length L are rotating with the same angular speed v (in rad/s) about axes that pass perpendicularly through one end. Rod A is massless but has a particle of mass 0.66 kg attached to its free end. Rod B has a mass of 0.66 kg, which is distributed uniformly along its length. The length of each rod is 0.75 m, and the angular speed is 4.2 rad/s. Find the kinetic energies of rod A with its attached particle and of rod B.

A fl ywheel is a solid disk that rotates about an axis that is perpendicular to the disk at its center. Rotating fl ywheels provide a means for storing energy in the form of rotational kinetic energy and are being considered as a possible alternative to batteries in electric cars. The gasoline burned in a 300-mile trip in a typical midsize car produces about 1.2 3 109 J of energy. How fast would a 13-kg fl ywheel with a radius of 0.30 m have to rotate to store this much energy? Give your answer in rev/min.

A helicopter has two blades (see Figure 8.11); each blade has a mass of 240 kg and can be approximated as a thin rod of length 6.7 m. The blades are rotating at an angular speed of 44 rad/s. (a) What is the total moment of inertia of the two blades about the axis of rotation? (b) Determine the rotational kinetic energy of the spinning blades.

A solid sphere is rolling on a surface. What fraction of its total kinetic energy is in the form of rotational kinetic energy about the center of mass?

Review Example 12 before attempting this problem. A marble and a cube are placed at the top of a ramp. Starting from rest at the same height, the marble rolls without slipping and the cube slides (no kinetic friction) down the ramp. Determine the ratio of the center-of-mass speed of the cube to the center-of-mass speed of the marble at the bottom of the ramp.

Starting from rest, a basketball rolls from the top of a hill to the bottom, reaching a translational speed of 6.6 m/s. Ignore frictional losses. (a) What is the height of the hill? (b) Released from rest at the same height, a can of frozen juice rolls to the bottom of the same hill. What is the translational speed of the frozen juice can when it reaches the bottom?

One end of a thin rod is attached to a pivot, about which it can rotate without friction. Air resistance is absent. The rod has a length of 0.80 m and is uniform. It is hanging vertically straight downward. The end of the rod nearest the fl oor is given a linear speed v0, so that the rod begins to rotate upward about the pivot. What must be the value of v0, such that the rod comes to a momentary halt in a straight-up orientation, exactly opposite to its initial orientation?

A bowling ball encounters a 0.760-m vertical rise on the way back to the ball rack, as the drawing illustrates. Ignore frictional losses and assume that the mass of the ball is distributed uniformly. The translational speed of the ball is 3.50 m/s at the bottom of the rise. Find the translational speed at the top

A tennis ball, starting from rest, rolls down the hill in the drawing. At the end of the hill the ball becomes airborne, leaving at an angle of 358 with respect to the ground. Treat the ball as a thin-walled spherical shell, and determine the range x.

Two disks are rotating about the same axis. Disk A has a moment of inertia of 3.4 kg ? m2 and an angular velocity of 17.2 rad/s. Disk B is rotating with an angular velocity of 29.8 rad/s. The two disks are then linked together without the aid of any external torques, so that they rotate as a single unit with an angular velocity of 22.4 rad/s. The axis of rotation for this unit is the same as that for the separate disks. What is the moment of inertia of disk B?

When some stars use up their fuel, they undergo a catastrophic explosion called a supernova. This explosion blows much or all of the stars mass outward, in the form of a rapidly expanding spherical shell. As a simple model of the supernova process, assume that the star is a solid sphere of radius R that is initially rotating at 2.0 revolutions per day After the star explodes, fi nd the angular velocity, in revolutions per day, of the expanding supernova shell when its radius is 4.0R. Assume that all of the stars original mass is contained in the shell.

Conceptual Example 13 provides useful background for this problem. A playground carousel is free to rotate about its center on frictionless bearings, and air resistance is negligible. The carousel itself (without riders) has a moment of inertia of 125 kg ? m2 . When one person is standing on the carousel at a distance of 1.50 m from the center, the carousel has an angular velocity of 0.600 rad/s. However, as this person moves inward to a point located 0.750 m from the center, the angular velocity increases to 0.800 rad/s. What is the persons mass?

Just after a motorcycle rides off the end of a ramp and launches into the air, its engine is turning counterclockwise at 7700 rev/min. The motorcycle rider forgets to throttle back, so the engines angular speed increases to 12 500 rev/min. As a result, the rest of the motorcycle (including the rider) begins to rotate clockwise about the engine at 3.8 rev/min. Calculate the ratio IE/IM of the moment of inertia of the engine to the moment of inertia of the rest of the motorcycle (and the rider). Ignore torques due to gravity and air resistance

A thin rod has a length of 0.25 m and rotates in a circle on a frictionless tabletop. The axis is perpendicular to the length of the rod at one of its ends. The rod has an angular velocity of 0.32 rad/s and a moment of inertia of 1.1 3 1023 kg ? m2 . A bug standing on the axis decides to crawl out to the other end of the rod. When the bug (mass 5 4.2 3 1023 kg) gets where its going, what is the angular velocity of the rod?

As seen from above, a playground carousel is rotating counterclockwise about its center on frictionless bearings. A person standing still on the ground grabs onto one of the bars on the carousel very close to its outer edge and climbs aboard. Thus, this person begins with an angular speed of zero and ends up with a nonzero angular speed, which means that he underwent a counterclockwise angular acceleration. The carousel has a radius of 1.50 m, an initial angular speed of 3.14 rad/s, and a moment of inertia of 125 kg ? m2 . The mass of the person is 40.0 kg. Find the fi nal angular speed of the carousel after the person climbs aboard.

A cylindrically shaped space station is rotating about the axis of the cylinder to create artifi cial gravity. The radius of the cylinder is 82.5 m. The moment of inertia of the station without people is 3.00 3 109 kg ? m2 . Suppose that 500 people, with an average mass of 70.0 kg each, live on this station. As they move radially from the outer surface of the cylinder toward the axis, the angular speed of the station changes. What is the maximum possible percentage change in the stations angular speed due to the radial movement of the people?

A thin, uniform rod is hinged at its midpoint. To begin with, onehalf of the rod is bent upward and is perpendicular to the other half. This bent object is rotating at an angular velocity of 9.0 rad/s about an axis that is perpendicular to the left end of the rod and parallel to the rods upward half (see the drawing). Without the aid of external torques, the rod suddenly assumes its straight shape. What is the angular velocity of the straight rod?

A small 0.500-kg object moves on a frictionless horizontal table in a circular path of radius 1.00 m. The angular speed is 6.28 rad/s. The object is attached to a string of negligible mass that passes through a small hole in the table at the center of the circle. Someone under the table begins to pull the string downward to make the circle smaller. If the string will tolerate a tension of no more than 105 N, what is the radius of the smallest possible circle on which the object can move?

A platform is rotating at an angular speed of 2.2 rad/s. A block is resting on this platform at a distance of 0.30 m from the axis. The coeffi - cient of static friction between the block and the platform is 0.75. Without any external torque acting on the system, the block is moved toward the axis. Ignore the moment of inertia of the platform and determine the smallest distance from the axis at which the block can be relocated and still remain in place as the platform rotates.

The drawing shows a lower leg being exercised. It has a 49-N weight attached to the foot and is extended at an angle u with respect to the vertical. Consider a rotational axis at the knee. (a) When u 5 90.08, fi nd the magnitude of the torque that the weight creates. (b) At what angle u does the magnitude of the torque equal 15 N ? m?

A solid disk rotates in the horizontal plane at an angular velocity of 0.067 rad/s with respect to an axis perpendicular to the disk at its center. The moment of inertia of the disk is 0.10 kg ? m2 . From above, sand is dropped straight down onto this rotating disk, so that a thin uniform ring of sand is formed at a distance of 0.40 m from the axis. The sand in the ring has a mass of 0.50 kg. After all the sand is in place, what is the angular velocity of the disk?

A solid cylindrical disk has a radius of 0.15 m. It is mounted to an axle that is perpendicular to the circular end of the disk at its center. When a 45-N force is applied tangentially to the disk, perpendicular to the radius, the disk acquires an angular acceleration of 120 rad/s2 . What is the mass of the disk?

Review Conceptual Example 7 before starting this problem. A uniform plank of length 5.0 m and weight 225 N rests horizontally on two supports, with 1.1 m of the plank hanging over the right support (see the drawing). To what distance x can a person who weighs 450 N walk on the overhanging part of the plank before it just begins to tip?

A rotating door is made from four rectangular sections, as indicated in the drawing. The mass of each section is 85 kg. A person pushes on the outer edge of one section with a force of F 5 68 N that is directed perpendicular to the section. Determine the magnitude of the doors angular acceleration

A block (mass 5 2.0 kg) is hanging from a massless cord that is wrapped around a pulley (moment of inertia 5 1.1 3 1023 kg ? m2 ), as the drawing shows. Initially the pulley is prevented from rotating and the block is stationary. Then, the pulley is allowed to rotate as the block falls. The cord does not slip relative to the pulley as the block falls. Assume that the radius of the cord around the pulley remains constant at a value of 0.040 m during the blocks descent. Find the angular acceleration of the pulley and the tension in the cord.

The drawing shows an outstretched arm (0.61 m in length) that is parallel to the fl oor. The arm is pulling downward against the ring attached to the pulley system, in order to hold the 98-N weight stationary. To pull the arm downward, the latissimus dorsi muscle applies the force M B in the drawing, at a point that is 0.069 m from the shoulder joint and oriented at an angle of 298. The arm has a weight of 47 N and a center of gravity (cg) that is located 0.28 m from the shoulder joint. Find the magnitude of M B.

A thin, rigid, uniform rod has a mass of 2.00 kg and a length of 2.00 m. (a) Find the moment of inertia of the rod relative to an axis that is perpendicular to the rod at one end. (b) Suppose all the mass of the rod were located at a single point. Determine the perpendicular distance of this point from the axis in part (a), such that this point particle has the same moment of inertia as the rod does. This distance is called the radius of gyration of the rod.

In outer space two identical space modules are joined together by a massless cable. These modules are rotating about their center of mass, which is at the center of the cable because the modules are identical (see the drawing). In each module, the cable is connected to a motor, so that the modules can pull each other together. The initial tangential speed of each module is v0 5 17 m/s. Then they pull together until the distance between them is reduced by a factor of two. Each module has a fi nal tangential speed of vf. Find the value of vf.

Two identical wheels are moving on horizontal surfaces. The center of mass of each has the same linear speed. However, one wheel is rolling, while the other is sliding on a frictionless surface without rolling. Each wheel then encounters an incline plane. One continues to roll up the incline, while the other continues to slide up. Eventually they come to a momentary halt, because the gravitational force slows them down. Each wheel is a disk of mass 2.0 kg. On the horizontal surfaces the center of mass of each wheel moves with a linear speed of 6.0 m/s. (a) What is the total kinetic energy of each wheel? (b) Determine the maximum height reached by each wheel as it moves up the incline.

An inverted V is made of uniform boards and weighs 356 N. Each side has the same length and makes a 30.08 angle with the vertical, as the drawing shows. Find the magnitude of the static frictional force that acts on the lower end of each leg of the V.

By means of a rope whose mass is negligible, two blocks are suspended over a pulley, as the drawing shows. The pulley can be treated as a uniform solid cylindrical disk. The downward acceleration of the 44.0-kg block is observed to be exactly one-half the acceleration due to gravity. Noting that the tension in the rope is not the same on each side of the pulley, fi nd the mass of the pulley.

A crate of mass 451 kg is being lifted by the mechanism shown in part a of the fi gure. The two cables are wrapped around their respective pulleys, which have radii of 0.600 and 0.200 m. The pulleys are fastened together to form a dual pulley and turn as a single unit about the center axle, relative to which the combined moment of inertia is 46.0 kg ? m2 . The cables roll on the dual pulley without slipping. A tension of magnitude 2150 N is maintained in the cable attached to the motor. Find the angular acceleration of the dual pulley and the tension in the cable attached to the crate.

The fi gure shows a uniform crate resting on a horizontal surface. The crate has a square cross section and a weight of W 5 580 N, which is uniformly distributed. At the bottom right edge of the surface is a small obstruction that prevents the crate from sliding when a horizontal pushing force P B is applied to the left side. However, if this force is great enough, the crate will begin to tip and rotate over the obstruction. Concepts: (i) What causes the tippingthe force P B or the torque that it creates? (ii) Where should P B be applied so that a minimum force will be necessary to provide the necessary torque? In other words, should the lever arm be a minimum or a maximum? Calculations: Determine the minimum pushing force that leads to tipping.

Two spheres are each rotating at an angular speed of 24 rad/s about axes that pass through their centers. Each has a radius of 0.20 m and a mass of 1.5 kg. However, as the fi gure shows, one is solid and the other is a thin-walled spherical shell. Suddenly, a net external torque due to friction (magnitude 5 0.12 N ? m) begins to act on each sphere and slows the motion down. Concepts: (i) Which sphere has the greater moment of inertia and why? (ii) Which sphere has the angular acceleration (a deceleration) with the smaller magnitude? (iii) Which sphere takes a longer time to come to a halt? Calculations: How long does it take each sphere to come to a halt?