Chapter 10

\(\cdot\) A cord is wrapped around the rim of a wheel 0.250 \(\mathrm{m}\) in radius, and a steady pull of 40.0 \(\mathrm{N}\) is exerted on the cord. The wheel is mounted on frictionless bearings on a horizontal shaft through its center. The moment of inertia of the wheel about this shaft is 5.00 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) Compute the angular acceleration of the wheel.

A certain type of propeller blade can be modeled as a thin uniform bar 2.50 m long and of mass 24.0 \(\mathrm{kg}\) that is free to rotate about a frictionless axle perpendicular to the bar at its midpoint. If a technician strikes this blade with a mallet 1.15 \(\mathrm{m}\) from the center with a 35.0 \(\mathrm{N}\) force perpendicular to the blade, find the maximum angular acceleration the blade could achieve.

\(\bullet\) A 750 gram grinding wheel 25.0 \(\mathrm{cm}\) in diameter is in the shape of a uniform solid disk. (We can ignore the small hole at the center.) When it is in use, it turns at a constant 220 \(\mathrm{rpm}\) about an axle perpendicular to its face through its center. When the power switch is turned off, you observe that the wheel stops in 45.0 s with constant angular acceleration due to friction at the axle. What torque does friction exert while this wheel is slowing down?

A solid, uniform cylinder with mass 8.25 \(\mathrm{kg}\) and diameter 15.0 \(\mathrm{cm}\) is spinning at 220 \(\mathrm{rpm}\) on a thin, frictionless axle that passes along the cylinder axis. You design a simple friction brake to stop the cylinder by pressing the brake against the outer rim with a normal force. The coefficient of kinetic friction between the brake and rim is \(0.333 .\) What must the applied normal force be to bring the cylinder to rest after it has turned through 5.25 revolutions?

A \(\mathrm{A} 2.00 \mathrm{kg}\) stone is tied to a thin, light wire wrapped around the outer edge of the uniform 10.0 \(\mathrm{kg}\) cylindrical pulley shown in Figure \(10.47 .\) The inner diameter of the pulley is \(60.0 \mathrm{cm},\) while the outer diameter is 1.00 \(\mathrm{m}\) . The system is released from rest, and there is no friction at the axle of the pulley. Find (a) the acceleration of the stone, (b) the tension in the wire, and (c) the angular acceleration of the pulley.

\bullet A \(22,500\) N elevator is to be accelerated upward by connecting it to a counterweight using a light (but strong!) cable passing over a solid uniform disk-shaped pulley. There is no appreciable friction at the axle of the pulley, but its mass is 875 \(\mathrm{kg}\) and it is 1.50 \(\mathrm{m}\) in diameter. (a) How heavy should the counterweight be so that it will accelerate the elevator upward through 6.75 \(\mathrm{m}\) in the first 3.00 \(\mathrm{s}\) , starting from rest? (b) Under these conditions, what is the tension in the cable on each side of the pulley?

\(\bullet\) A thin, light string is wrapped around the rim of a 4.00 kg solid uniform disk that is 30.0 \(\mathrm{cm}\) in diameter. A person pulls on the string with a constant force of 100.0 \(\mathrm{N}\) tangent to the disk, as shown in Figure \(10.49 .\) The disk is not attached to anything and is free to move and tum. (a) Find the angular acceleration of the disk about its center of mass and the linear acceleration of its center of mass. (b) If the disk is replaced by a hollow thin-walled cylinder of the same mass and diameter, what will be the accelerations in part (a)?

A hollow spherical shell with mass 2.00 \(\mathrm{kg}\) rolls without slipping down a \(38.0^{\circ}\) slope. (a) Find the acceleration of the shell and the friction force on it. Is the friction kinetic or static friction? Why? (b) How would your answers to part (a) change if the mass were doubled to 4.00 \(\mathrm{kg} ?\)

A solid disk of radius 8.50 \(\mathrm{cm}\) and mass \(1.25 \mathrm{kg},\) which is rolling at a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) , begins rolling without slipping up a \(10.0^{\circ}\) slope. How long will it take for the disk to come to a stop?

\(\cdot\) What is the power output in horsepower of an electric motor turning at 4800 rev/min and developing a torque of 4.30 \(\mathrm{N} \cdot \mathrm{m}\) ?

A playground merry-go-round has a radius of 4.40 \(\mathrm{m}\) and a moment of inertia of 245 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) and turns with negligible friction about a vertical axle through its center. (a) A child applies a 25.0 \(\mathrm{N}\) force tangentially to the edge of the merry-go-round for 20.0 s. If the merry-go-round is initially at rest, what is its angular velocity after this 20.0 s interval? (b) How much work did the child do on the merry-go- round? (c) What is the average power supplied by the child?

\(\cdot\) The flywheel of a motor has a mass of 300.0 \(\mathrm{kg}\) and a moment of inertia of 580 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) The motor develops a constant torque of \(2000.0 \mathrm{N} \cdot \mathrm{m},\) and the flywheel starts from rest. (a) What is the angular acceleration of the flywheel? (b) What is its angular velocity after it makes 4.00 revolutions? (c) How much work is done by the motor during the first 4.00 revolutions?

(a) Compute the torque developed by an industrial motor whose output is 150 \(\mathrm{kW}\) at an angular speed of 4000.0 rev/min. (b) A drum with negligible mass and 0.400 \(\mathrm{m}\) in diameter is attached to the motor shaft, and the power output of the motor is used to raise a weight hanging from a rope wrapped around the drum. How heavy a weight can the motor lift at constant speed? (c) At what constant speed will the weight rise?

\(\bullet\) Calculate the angular momentum and kinetic energy of a solid uniform sphere with a radius of 0.120 \(\mathrm{m}\) and a mass of 14.0 \(\mathrm{kg}\) if it is rotating at 6.00 \(\mathrm{rad} / \mathrm{s}\) about an axis through its center.

A small 4.0 kg brick is released from rest 2.5 \(\mathrm{m}\) above a horizontal seesaw on a fulcrum at its center, as shown in Figure 10.52 . Find the angular momentum of this brick about a horizontal axis through the fulcrum and perpendicular to the plane of the figure (a) the instant the brick is released and (b) the instant before it strikes the seesaw.

\(\cdot\) A woman with mass 50.0 \(\mathrm{kg}\) is standing on the rim of a large disk that is rotating at 0.50 \(\mathrm{rev} / \mathrm{s}\) about an axis perpendicular to it through its center. The disk has a mass of 110 \(\mathrm{kg}\) and a radius of 4.0 \(\mathrm{m}\) . Calculate the magnitude of the total angular momentum of the woman-plus-disk system, assuming that you can treat the woman as a point.

\(\bullet\) A certain drawbridge can be modeled as a uniform \(15,000 \mathrm{N}\) bar, 12.0 \(\mathrm{m}\) long, pivoted about its lower end. When this bridge is raised to an angle of \(60.0^{\circ}\) above the horizontal, the cable holding it suddenly breaks, allowing the bridge to fall. At the instant after the cable breaks, (a) what is the torque on this bridge about the pivot and (b) at what rate is its angular momentum changing?

\(\cdot\) On an old-fashioned rotating piano stool, a woman sits holding a pair of dumbbells at a distance of 0.60 m from the axis of rotation of the stool. She is given an angular velocity of 3.00 rad/s, after which she pulls the dumbbells in until they are only 0.20 m distant from the axis. The woman's moment of inertia about the axis of rotation is 5.00 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) and may be considered constant. Each dumbbell has a mass of 5.00 \(\mathrm{kg}\) and may be considered a point mass. Neglect friction. (a) What is the initial angular momentum of the system? (b) What is the angular velocity of the system after the dumbbells are pulled in toward the axis? (c) Compute the kinetic energy of the system before and after the dumbbells are pulled in. Account for the difference, if any.

\(\cdot\) The spinning figure skater. The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center. (See Figure \(10.53 .\) ) When the skater's hands and arms are brought in and wrapped around his body to execute the spin, the hands and arms can be considered a thin-walled hollow cylinder. His hands and arms have a combined mass of 8.0 \(\mathrm{kg}\) . When outstretched, they span 1.8 \(\mathrm{m}\) ; when wrapped, they form a cylinder of radius 25 \(\mathrm{cm} .\) The moment of inertia about the axis of rotation of the remainder of his body is constant and equal to 0.40 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) If the skater's original angular speed is 0.40 \(\mathrm{rev} / \mathrm{s}\) what is his final angular speed?

\bullet A small block on a frictionless horizontal surface has a mass of 0.0250 \(\mathrm{kg}\) . It is attached to a massless cord passing through a hole in the surface. (See Figure \(10.54 .\) The block is originally revolving at a distance of 0.300 \(\mathrm{m}\) from the hole with an angular speed of 1.75 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.150 \(\mathrm{m} .\) You may treat the block as a particle. (a) Is angular momentum conserved? Why or why not? (b) What is the new angular speed? (c) Find the change in kinetic energy of the block. (d) How much work was done in pulling the cord?

. A uniform 4.5 \(\mathrm{kg}\) square solid wooden gate 1.5 \(\mathrm{m}\) on each side hangs vertically from a frictionless pivot at the center of its upper edge. A 1.1 \(\mathrm{kg}\) raven flying horizontally at 5.0 \(\mathrm{m} / \mathrm{s}\) flies into this gate at its center and bounces back at 2.0 \(\mathrm{m} / \mathrm{s}\) in the opposite direction. (a) What is the angular speed of the gate just after it is struck by the unfortunate raven? (b) During the collision, why is the angular momentum conserved, but not the linear momentum?

A diver comes off a board with arms straight up and legs straight down, giving her a moment of inertia about her rotation axis of 18 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) She then tucks into a small ball, decreasing this moment of inertia to 3.6 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) . While tucked, she makes two complete revolutions in 1.0 s. If she hadn't tucked at all, how many revolutions would she have made in the 1.5 s from board to water?

A A large turntable rotates about a fixed vertical axis, making one revolution in 6.00 s. The moment of inertia of the turntable about this axis is 1200 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) A child of mass \(40.0 \mathrm{kg},\) initially standing at the center of the turntable, runs out along a radius. What is the angular speed of the turntable when the child is 2.00 \(\mathrm{m}\) from the center, assuming that you can treat the child as a particle?

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

Supporting a broken leg. A therapist tells a 74 kg patient with a broken leg that he must have his leg in a cast suspended horizontally. For minimum discomfort, the leg should be supported by a vertical strap attached at the center of mass of the leg-cast system. (See Figure \(10.57 . )\) In order to comply with these instructions, the patient consults a table of typical mass distributions and finds that both upper legs (thighs) together typically account for 21.5\(\%\) of body weight and the center of mass of each thigh is 18.0 \(\mathrm{cm}\) from the hip joint. The patient also reads that two lower legs (including the feet) are 14.0\(\%\) of body weight, with a center of mass 69.0 \(\mathrm{cm}\) from the hip joint. The cast has a mass of \(5.50 \mathrm{kg},\) and its center of mass is 78.0 \(\mathrm{cm}\) from the hip joint. How far from the hip joint should the supporting strap be attached to the cast?

Two people are carrying a uniform wooden board that is 3.00 \(\mathrm{m}\) long and weighs 160 \(\mathrm{N}\) . If one person applies an upward force equal to 60 \(\mathrm{N}\) at one end, at what point and with what force does the other person lift? Start with a free-body diagram of the board.

\(\bullet\) Two people carry a heavy electric motor by placing it on a light board 2.00 m long. One person lifts at one end with a force of \(400.0 \mathrm{N},\) and the other lifts at the opposite end with a force of 600.0 \(\mathrm{N}\) . (a) Start by making a free-body diagram of the motor. (b) What is the weight of the motor? (c) Where along the board is its center of gravity located?

A uniform 250 \(\mathrm{N}\) ladder rests against a perfectly smooth wall, making a \(35^{\circ}\) angle with the wall. (a) Draw a free-body diagram of the ladder. (b) Find the normal forces that the wall and the floor exert on the ladder. (c) What is the friction force on the ladder at the floor?

Two people carry a heavy electric motor by placing it on a light board 2.00 m long. One person lifts at one end with a force of \(400 \mathrm{N},\) and the other lifts the opposite end with a force of 600 \(\mathrm{N}\) . (a) What is the weight of the motor, and where along the board is its center of gravity located? (b) Suppose the board is not light but weighs \(200 \mathrm{N},\) with its center of gravity at its center, and the two people each exert the same forces as before. What is the weight of the motor in this case, and where is its center of gravity located?

\(\bullet\) A uniform, \(90.0-\mathrm{N}\) table is 3.6 \(\mathrm{m}\) long, 1.0 \(\mathrm{m}\) high, and 1.2 \(\mathrm{m}\) wide. \(\mathrm{A} 1500-\mathrm{N}\) weight is placed 0.50 \(\mathrm{m}\) from one end of the table, a distance of 0.60 \(\mathrm{m}\) from each of the two legs at that end. Draw a free-body diagram for the table and find the force that each of the four legs exerts on the floor.

\(\bullet\) For each of the following rotating objects, describe the direction of the angular momentum vector: (a) the minute hand of a clock; (b) the right front tire of a car moving backwards; (c) an ice skater spinning clockwise; (d) the earth, rotating on its axis.

Back pains during pregnancy. Women often suffer from back pains during pregnancy. Let us investigate the cause of these pains, assuming that the woman's mass is 60 kg before pregnancy. Typically, women gain about 10 kg during pregnancy, due to the weight of the fetus, placenta, amniotic fluid, etc. To make the calculations easy, but still realistic, we shall model the unpregnant woman as a uniform cylinder of diameter 30 \(\mathrm{cm} .\) We can model the added mass due to the fetus as a 10 kg sphere 25 \(\mathrm{cm}\) in diameter and centered about 5 \(\mathrm{cm}\) outside the woman's original front surface. (a) By how much does her pregnancy change the horizontal location of the woman's center of mass? (b) How does the change in part (a) affect the way the pregnant woman must stand and walk? In other words, what must she do to her posture to make up for her shifted center of mass? (c) Can you now explain why she might have backaches?

Prior to being placed in its hole, a \(5700-\mathrm{N}, 9.0\) -m-long, uniform utility pole makes some nonzero angle with the vertical. A vertical cable attached 2.0 \(\mathrm{m}\) below its upper end holds it in place while its lower end rests on the ground. (a) Find the tension in the cable and the magnitude and direction of the force exerted by the ground on the pole. (b) Why don't we need to know the angle the pole makes with the vertical, as long as it is not zero?

A a uniform drawbridge must be held at a \(37^{\circ}\) angle above the horizontal to allow ships to pass underneath. The drawbridge weighs \(45,000 \mathrm{N},\) is 14.0 \(\mathrm{m}\) long, and pivots about a hinge at its lower end. A cable is connected 3.5 \(\mathrm{m}\) from the hinge, as measured along the bridge, and pulls horizontally on the bridge to hold it in place. (a) What is the tension in the cable? (b) Find the magnitude and direction of the force the hinge exerts on the bridge. (c) If the cable suddenly breaks, what is the initial angular acceleration of the bridge?

Atwood's machine. Figure 10.73 illustrates an Atwood's machine. Find the linear accelerations of blocks \(A\) and \(B,\) the angular acceleration of the wheel \(C,\) and the tension in each side of the cord if there is no slipping between the cord and the surface of the wheel. Let the masses of blocks \(A\) and \(B\) be 4.00 \(\mathrm{kg}\) and 2.00 \(\mathrm{kg}\) , respectively, the moment of inertia of the wheel about its axis be \(0.300 \mathrm{kg} \cdot \mathrm{m}^{2},\) and the radius of the wheel be 0.120 \(\mathrm{m} .\)

Supporting an injured arm: I. A 650 N person must have her injured arm supported, with the upper arm horizontal and the fore- arm vertical. (See Figure \(10.76 .\) ) According to biomedical tables and direct measurements, her upper arm is 26 \(\mathrm{cm}\) long (measured from the shoulder joint), accounts for 3.50\(\%\) of her body weight, and has a center of mass 13.0 \(\mathrm{cm}\) from her shoulder joint. Her forearm (including the hand) is 34.0 \(\mathrm{cm}\) long, makes up 3.25\(\%\) of her body weight, and has a center of mass 43.0 \(\mathrm{cm}\) from her shoulder joint. (a) Where is the center of mass of the person's arm when it is supported as shown? (b) What weight \(W\) is needed to support her arm? (c) Find the horizontal and vertical components of the force that the shoulder joint exerts on her arm.

You are trying to raise a bicycle wheel of mass \(m\) and radius \(R\) up over a curb of height \(h .\) To do this, you apply a horizontal force \(\vec{F}\) (Fig. \(10.81 ) .\) What is the smallest magnitude of the force \(\vec{F}\) that will succeed in raising the wheel onto the curb when the force is applied (a) at the center of the wheel, and (b) at the top of the wheel? (c) In which case is less force required?

An experimental bicycle wheel is placed on a test stand so that it is free to turn on its axle. If a constant net torque of 5.00 \(\mathrm{N} \cdot \mathrm{m}\) is applied to the tire for 2.00 \(\mathrm{s}\) , the angular speed of the tire increases from zero to 100 rev/min. The external torque is then removed, and the wheel is brought to rest in 125 s by friction in its bearings. Compute (a) the moment of inertia of the wheel about the axis of rotation, (b) the friction torque, and (c) the total number of revolutions made by the wheel in the 125 s time interval.

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

Disks \(A\) and \(B\) are mounted on shaft \(S S\) and may be connected or dis- connected by clutch \(C .\) (See Figure \(10.82 . )\) Disk \(A\) is made of a lighter material than disk \(B,\) so the moment of inertia of disk \(A\) about the shaft is one-third that of disk \(B\) . The moments of inertia of the shaft and clutch are negligible. With the clutch disconnected, \(A\) is brought up to an angular speed \(\omega_{0 . \text { . The }}\) accelerating torque is then removed from \(A,\) and \(A\) , and \(A\) is coupled to disk \(B\) by the clutch. (You can ignore bearing friction.) It is found that 2400 \(\mathrm{J}\) of thermal energy is developed in the clutch when the connection is made. What was the original kinetic energy of disk \(A\) ?

While exploring a castle, Exena the Exterminator is spotted by a dragon who chases her down a hallway. Exena runs into a room and attempts to swing the heavy door shut before the dragon gets her. The door is initially perpendicular to the wall, so it must be turned through \(90^{\circ}\) to close. The door is 3.00 \(\mathrm{m}\) tall and 1.25 \(\mathrm{m}\) wide, and it weighs 750 \(\mathrm{N} .\) You can ignore the friction at the hinges. If Exena applies a force of 220 \(\mathrm{N}\) at the edge of the door and perpendicular to it, how much time does it take her to close the door?

What would be the radial (centripetal) acceleration of an astronaut standing on the inner surface of the space station, which is at a distance of 240 \(\mathrm{m}\) from the axis of rotation, once the space station reaches its final angular velocity of 0.20 \(\mathrm{rad} / \mathrm{s}\) ? A. 0 \(\mathrm{m} / \mathrm{s}^{2}\) B. 4.8 \(\mathrm{m} / \mathrm{s}^{2}\) \(\mathrm{C}, 9.6 \mathrm{m} / \mathrm{s}^{2}\) D. 48 \(\mathrm{m} / \mathrm{s}^{2}\)