Chapter 10

One force acting on a machine part is \(\vec{\boldsymbol{F}}=(-5.00 \mathrm{N}) \hat{\imath}+\) \((4.00 \mathrm{N}) \hat{\boldsymbol{J}}\) . The vector from the origin to the point where the force is applied is \(\vec{r}=(-0.450 \mathrm{m}) \hat{\imath}+(0.150 \mathrm{m}) \hat{\boldsymbol{J}}\) (a) In a sketch, show \(\vec{r}, \vec{\boldsymbol{F}},\) and the origin. (b) Use the right-hand rule to determine the direction of the torque. (c) Calculate the vector torque for an axis at the origin produced by this force. Verify that the direction of the torque is the same as you obtained in part (b).

A metal bar is in the \(x y-\) plane with one end of the bar at the origin. A force \(\vec{F}=(7.00 \mathrm{N}) \hat{\imath}+(-3.00 \mathrm{N}) \hat{J}\) is applied to the bar at the point \(x=3.00 \mathrm{m}, y=4.00 \mathrm{m} .\) (a) In terms of unit vectors \(\hat{\boldsymbol{i}}\) and \(\hat{\boldsymbol{J}},\) what is the position vector \(\vec{r}\) for the point where the force is applied? (b) What are the magnitude and direction of the torque with respect to the origin produced by \(\vec{\boldsymbol{F}}\) ?

The flywheel of an engine has moment of inertia 2.50 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) about its rotation axis. What constant torque is required to bring it up to an angular speed of 400 \(\mathrm{rev} / \mathrm{min}\) in 8.00 \(\mathrm{s}\) starting from rest?

A uniform disk with mass 40.0 \(\mathrm{kg}\) and radius 0.200 \(\mathrm{m}\) is pivoted at its center about a horizontal, frictionless axle that is stationary. The disk is initially at rest, and then a constant force \(F=30.0 \mathrm{N}\) is applied tangent to the rim of the disk. (a) What is the magnitude \(v\) of the tangential velocity of a point on the rim of the disk after the disk has turned through 0.200 revolution? (b) What is the magnitude \(a\) of the resultant acceleration of a point on the rim of the disk after the disk has turned through 0.200 revolution?

A machine part has the shape of a solid uniform sphere of mass 225 \(\mathrm{g}\) and diameter 3.00 \(\mathrm{cm} .\) It is spinning about a frictionless axle through its center, but at one point on its equator it is scraping against metal, resulting in a friction force of 0.0200 \(\mathrm{N}\) at that point. (a) Find its angular acceleration. (b) How long will it take to decrease its rotational speed by 22.5 \(\mathrm{rad} / \mathrm{s} ?\)

A cord is wrapped around the rim of a solid uniform wheel 0.250 \(\mathrm{m}\) in radius and of mass 9.20 \(\mathrm{kg} .\) A steady horizontal pull of 40.0 \(\mathrm{N}\) to the right is exerted on the cord, pulling it off tangentially from the wheel. The wheel is mounted on frictionless bearings on a horizontal axle through its center. (a) Compute the angular acceleration of the wheel and the acceleration of the part of the cord that has already been pulled off the wheel. (b) Find the magnitude and direction of the force that the axle exerts on the wheel. (c) Which of the answers in parts (a) and (b) would change if the pull were upward instead of horizontal?

A 2.00 -kg textbook rests on a frictionless, horizontal surface. A cord attached to the book passes over a pulley whose diameter is \(0.150 \mathrm{m},\) to a hanging book with mass 3.00 \(\mathrm{kg}\) . The system is released from rest, and the books are observed to move 1.20 \(\mathrm{m}\) in 0.800 s. (a) What is the tension in each part of the cord? (b) What is the moment of inertia of the pulley about its rotation axis?

A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, similar to what is shown in Fig. \(10.10 .\) The pulley is a uniform disk with mass 10.0 \(\mathrm{kg}\) and radius 50.0 \(\mathrm{cm}\) and turns on frictionless bearings. You measure that the stone travels 12.6 \(\mathrm{m}\) in the first 3.00 s starting from rest. Find (a) the mass of the stone and (b) the tension in the wire.

A wheel rotates without friction about a stationary horizontal axis at the center of the wheel. A constant tangential force equal to 80.0 \(\mathrm{N}\) is applied to the rim of the wheel. The wheel has radius 0.120 \(\mathrm{m} .\) Starting from rest, the wheel has an angular speed of 12.0 rev/s after 2.00 s. What is the moment of inertia of the wheel?

A 15.0 -kg bucket of water is suspended by a very light rope wrapped around a solid uniform cylinder 0.300 \(\mathrm{m}\) in diameter with mass 12.0 \(\mathrm{kg} .\) The cylinder pivots on a frictionless axle through its center. The bucket is released from rest at the top of a well and falls 10.0 \(\mathrm{m}\) to the water. (a) What is the tension in the rope while the bucket is falling? (b) With what speed does the bucket strike the water? (c) What is the time of fall? (d) While the bucket is falling, what is the force exerted on the cylinder by the axle?

BIO Gymnastics. We can roughly model a gymnastic tumbler as a uniform solid cylinder of mass 75 \(\mathrm{kg}\) and diameter 1.0 \(\mathrm{m}\) . If this tumbler rolls forward at 0.50 rev/s, (a) how much total kinetic energy does he have, and (b) what percent of his total kinetic energy is rotational?

A \(2.20-\) kg hoop 1.20 \(\mathrm{m}\) in diameter is rolling to the right without slipping on a horizontal floor at a steady 3.00 \(\mathrm{rad} / \mathrm{s}\) . (a) How fast is its center moving? (b) What is the total kinetic energy of the hoop? (c) Find the velocity vector of each of the following points, as viewed by a person at rest on the ground: (i) the highest point on the hoop; (ii) the lowest point on the hoop; (iii) a point on the right side of the hoop, midway between the top and the bottom. (d) Find the velocity vector for each of the points in part (c), but this time as viewed by someone moving along with the same velocity as the hoop.

A string is wrapped several times around the rim of a small hoop with radius 8.00 \(\mathrm{cm}\) and mass 0.180 \(\mathrm{kg} .\) The free end of the string is held in place and the hoop is released from rest (Fig. E10.20). After the hoop has descended \(75.0 \mathrm{cm},\) calculate (a) the angular speed of the rotating hoop and (b) the speed of its center.

What fraction of the total kinetic energy is rotational for the following objects rolling without slipping on a horizontal surface? (a) a uniform solid cylinder; (b) a uniform sphere; ( c) a thin walled, hollow sphere; (d) a hollow cylinder with outer radius \(R\) and inner radius \(R / 2\) .

A hollow, spherical shell with mass 2.00 \(\mathrm{kg}\) rolls without slipping down a \(38.0^{\circ}\) slope. (a) Find the acceleration, the friction force, and the minimum coefficient of friction needed to prevent slipping. (b) How would your answers to part (a) change if the mass were doubled to 4.00 \(\mathrm{kg}\) ?

A uniform marble rolls down a symmetrical bowl, starting from rest at the top of the left side. The top of each side is a distance \(h\) above the bottom of the bowl. The left half of the bowl is rough enough to cause the marble to roll without slipping, but the right half has no friction because it is coated with oil. (a) How far up the smooth side will the marble go, measured vertically from the bottom? (b) How high would the marble go if both sides were as rough as the left side? (c) How do you account for the fact that the marble goes higher with friction on the right side than without friction?

A \(392-\mathrm{N}\) wheel comes off a moving truck and rolls with- out slipping along a highway. At the bottom of a hill it is rotating at 25.0 \(\mathrm{rad} / \mathrm{s} .\) The radius of the wheel is \(0.600 \mathrm{m},\) and its moment of inertia about its rotation axis is 0.800\(M R^{2} .\) Friction does work on the wheel as it rolls up the hill to a stop, a height \(h\) above the bottom of the hill; this work has absolute value 3500 \(\mathrm{J} .\) Calculate \(h .\)

A bicycle racer is going downhill at 11.0 \(\mathrm{m} / \mathrm{s}\) when, to his horror, one of his \(2.25-\mathrm{kg}\) wheels comes off as he is 75.0 \(\mathrm{m}\) above the foot of the hill. We can model the wheel as a thin- walled cylinder 85.0 \(\mathrm{cm}\) in diameter and neglect the small mass of the spokes. (a) How fast is the wheel moving when it reaches the foot of the hill if it rolled without slipping all the way down? (b) How much total kinetic energy does the wheel have when it reaches the bottomof the hill?

A size-5 soccer ball of diameter 22.6 \(\mathrm{cm}\) and mass 426 \(\mathrm{g}\) rolls up a hill without slipping, reaching a maximum height of 5.00 \(\mathrm{m}\) above the base of the hill. We can model this ball as a thin- walled hollow sphere. (a) At what rate was it rotating at the base of the hill? (b) How much rotational kinetic energy did it have then?

An engine delivers 175 hp to an aircraft propeller at 2400 rev/min. (a) How much torque does the aircraft engine provide? (b) How much work does the engine do in one revolution of the propeller?

A playground merry-go-round has radius 2.40 \(\mathrm{m}\) and moment of inertia 2100 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) about a vertical axle through its center, and it turns with negligible friction. (a) A child applies an 18.0 -N force tangentially to the edge of the merry-go-round for 15.0 s. If the merry- go-round is initially at rest, what is its angular speed after this 15.0 -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?

An electric motor consumes 9.00 \(\mathrm{kJ}\) of electrical energy in 1.00 \(\mathrm{min} .\) If one-third of this energy goes into heat and other forms of internal energy of the motor, with the rest going to the motor output, how much torque will this engine develop if you run it at 2500 rpm?

(a) Compute the torque developed by an industrial motor whose output is 150 \(\mathrm{kW}\) at an angular speed of 4000 \(\mathrm{rev} / \mathrm{min}\) . (b) A drum with negligible mass, 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?

A woman with mass 50 \(\mathrm{kg}\) is standing on the rim of a large disk that is rotating at 0.50 \(\mathrm{rev} / \mathrm{s}\) about an axis through its center. The disk has mass 110 \(\mathrm{kg}\) and radius 4.0 \(\mathrm{m} .\) Calculate the magnitude of the total angular momentum of the woman-disk system. (Assume that you can treat the woman as a point.)

Find the magnitude of the angular momentum of the second hand on a clock about an axis through the center of the clock face. The clock hand has a length of 15.0 \(\mathrm{cm}\) and a mass of 6.00 g. Take the second hand to be a slender rod rotating with constant angular velocity about one end.

A hollow, thin-walled sphere of mass 12.0 \(\mathrm{kg}\) and diameter 48.0 \(\mathrm{cm}\) is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by \(\theta(t)=A t^{2}+B t^{4},\) where \(A\) has numerical value 1.50 and \(B\) has numerical value 1.10 . (a) What are the units of the constants \(A\) and \(B ?\) (b) At the time 3.00 s, find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.

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 sphere, 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.

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 (Fig. E10.42). The block is originally revolving at a distance of 0.300 m from the hole with an angular speed of 1.75 \(\mathrm{rad} / \mathrm{s}\) . The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.150 \(\mathrm{m} .\) Model the block as a particle. (a) Is the angular momentum of the block 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?

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 (Fig. Elo.43). 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 rotation axis of the remainder of his body is constant and equal to 0.40 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) . If his original angular speed is \(0.40 \mathrm{rev} / \mathrm{s},\) what is his final angular speed?

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 \(\mathrm{s}\) 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 large wooden turntable in the shape of a flat uniform 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 near the outer edge. (a) Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.) (b) Compute the kinetic energy of the system before and after the parachutist lands. Why are these kinetic energies not equal?

A solid wood door 1.00 \(\mathrm{m}\) wide and 2.00 \(\mathrm{m}\) high is hinged along one side and has a total mass of 40.0 \(\mathrm{kg}\) . Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass 0.500 \(\mathrm{kg}\) , traveling perpendicular to the door at 12.0 \(\mathrm{m} / \mathrm{s}\) just before impact. Find the final angular speed of the door. Does the mud make a significant contribution to the moment of inertia?

A small 10.0-g bug stands at one end of a thin uniform bar that is initially at rest on a smooth horizontal table. The other end of the bar pivots about a nail driven into the table and can rotate freely, without friction. The bar has mass 50.0 \(\mathrm{g}\) and is 100 \(\mathrm{cm}\) in length. The bug jumps off in the horizontal direction, perpendicular to the bar, with a speed of 20.0 \(\mathrm{cm} / \mathrm{s}\) relative to the table. (a) What is the angular speed of the bar just after the frisky insect leaps? (b) What is the total kinetic energy of the system just after the bug leaps? (c) Where does this energy come from?

Asteroid Collision! Suppose that an asteroid traveling straight toward the center of the earth were to collide with our planet at the equator and bury itself just below the surface. What would have to be the mass of this asteroid, in terms of the earth's mass \(M\) , for the day to become 25.0\(\%\) longer than it presently is as a result of the collision? Assume that the asteroid is very small compared to the earth and that the earth is uniform throughout.

A thin, uniform metal bar, 2.00 \(\mathrm{m}\) long and weighing \(90.0 \mathrm{N},\) is hanging vertically from the ceiling by a frictionless pivot. Suddenly it is struck 1.50 \(\mathrm{m}\) below the ceiling by a small 3.00 -kg ball, initially traveling horizontally at 10.0 \(\mathrm{m} / \mathrm{s} .\) The ball rebounds in the opposite direction with a speed of 6.00 \(\mathrm{m} / \mathrm{s}\) . (a) Find the angular speed of the bar just after the collision. (b) During the collision, why is the angular momentum conserved but not the linear momentum?

A thin uniform rod has a length of 0.500 \(\mathrm{m}\) and is rotating in a circle on a frictionless table. The axis of rotation is perpendicular to the length of the rod at one end and is stationary. The rod has an angular velocity of 0.400 \(\mathrm{rad} / \mathrm{s}\) and a moment of inertia about the axis of \(3.00 \times 10^{-3} \mathrm{kg} \cdot \mathrm{m}^{2} .\) A bug initially standing on the rod at the axis of rotation decides to crawl out to the other end of the rod. When the bug has reached the end of the rod and sits there, its tangential speed is 0.160 \(\mathrm{m} / \mathrm{s}\) . The bug can be treated as a point mass. (a) What is the mass of the rod? (b) What is the mass of the bug?

A uniform, 4.5 -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 door 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?

Sedna. In November \(2003,\) the now-most-distant-known object in the solar system was discovered by observation with a telescope on Mt. Palomar. This object, known as Sedna, is approximately 1700 \(\mathrm{km}\) in diameter, takes about \(10,500\) years to orbit our sun, and reaches a maximum speed of 4.64 \(\mathrm{km} / \mathrm{s}\) . Calculations of its complete path, based on several measurements of its position, indicate that its orbit is highly elliptical, varying from 76 \(\mathrm{AU}\) to 942 \(\mathrm{AU}\) in its distance from the sun, where \(\mathrm{AU}\) is the astronomical unit, which is the average distance of the earth from the sun \(\left(1.50 \times 10^{8} \mathrm{km}\right)\) . (a) What is Sedna's minimum speed? (b) At what points in its orbit do its maximum and minimum speeds occur? (c) What is the ratio of Sedna's maximum kinetic energy to its minimum kinetic energy?

The rotor (flywheel) of a toy gyroscope has mass 0.140 kg. Its moment of inertia about its axis is \(1.20 \times 10^{-4} \mathrm{kg} \cdot \mathrm{m}^{2} .\) The mass of the frame is 0.0250 \(\mathrm{kg}\) . The gyroscope is supported on a single pivot (Fig. E10.53) with its center of mass a horizontal distance of 4.00 \(\mathrm{cm}\) from the pivot. The gyroscope is precessing in a horizontal plane at the rate of one revolution in 2.20 \(\mathrm{s}\) . (a) Find the upward force exerted by the pivot. (b) Find the angular speed with which the rotor is spinning about its axis, expressed in rev/min. (c) Copy the diagram and draw vectors to show the angular momentum of the rotor and the torque acting on it.

A Gyroscope on the Moon. A certain gyroscope precesses at a rate of 0.50 \(\mathrm{rad} / \mathrm{s}\) when used on earth. If it were taken to a lunar base, where the acceleration due to gravity is \(0.165 g,\) what would be its precession rate?

A gyroscope is precessing about a vertical axis. Describe what happens to the precession angular speed if the following changes in the variables are made, with all other variables remaining the same: (a) the angular speed of the spinning flywheel is doubled; (b) the total weight is doubled; (c) the moment of inertia about the axis of the spinning flywheel is doubled; (d) the distance from the pivot to the center of gravity is doubled. (e) What happens if all four of the variables in parts (a) through (d) are doubled?

A \(50.0\)-kg grindstone is a solid disk 0.520 \(\mathrm{m}\) in diameter. You press an ax down on the rim with a normal force of 160 \(\mathrm{N}\) (Fig. P10.57). The coefficient of kinetion between the blade and the stone is 0.60 , and there is a constant friction torque of 6.50 \(\mathrm{N} \cdot \mathrm{m}\) between the axle of the stone and its bearings. (a) How much force must be applied tangentially at the end of a crank handle 0.500 \(\mathrm{m}\) long to bring the stone from rest to 120 \(\mathrm{rev} / \mathrm{min}\) in 9.00 \(\mathrm{s} ?\) (b) After the grindstone attains an angular speed of 120 \(\mathrm{rev} / \mathrm{min}\) , what tangential force at the end of the handle is needed to maintain a constant angular speed of 120 \(\mathrm{rev} / \mathrm{min}\) ? (c) How much time does it take the grindstone to come from 120 \(\mathrm{rev} / \mathrm{min}\) to rest if it is acted on by the axle friction alone?

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 7.00 \(\mathrm{N} \cdot \mathrm{m}\) is applied to the tire for 2.00 \(\mathrm{s}\) , the angular speed of the tire increases from 0 to 100 rev/min. The external torque is then removed, and the wheel is brought to rest by friction in its bearings in 125 s. Compute (a) the moment of inertia of the wheel about the rotation axis; (b) the friction torque; (c) the total number of revolutions made by the wheel in the 125 -s time interval.

A grindstone in the shape of a solid disk with diameter 0.520 \(\mathrm{m}\) and a mass of 50.0 \(\mathrm{kg}\) is rotating at 850 rev/min. You press an ax against the rim with a normal force of 160 \(\mathrm{N}\) (Fig. P10.57), and the grindstone comes to rest in 7.50 s. Find the coefficient of friction between the ax and the grindstone. You can ignore friction in the bearings.

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 uniform hollow disk has two pieces of thin, light wire wrapped around its outer rim and is supported from the ceiling (Fig. P10.62). Suddenly one of the wires breaks, and the remaining wire does not slip as the disk rolls down. Use energy conservation to find the speed of the center of this disk after it has fallen a distance of 2.20 \(\mathrm{m} .\)

A thin, uniform, \(3.80-\mathrm{kg}\) bar, 80.0 \(\mathrm{cm}\) long, has very small \(2.50-\mathrm{kg}\) balls glued on at either end (Fig. P10.63). It is supported horizontally by a thin, horizontal, frictionless axle passing through its center and perpendicular to the bar. Suddenly the right-hand ball becomes detached and falls off, but the other ball remains glued to the bar. (a) Find the angular acceleration of the bar just after the ball falls off. (b) Will the angular acceleration remain constant as the bar continues to swing? If not, will it increase or decrease? (c) Find the angular velocity of the bar just as it swings through its vertical position.

While exploring a castle, Exena the Exterminator is spotted by a dragon that 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?

Atwood's Machine. Figure Pl0.67 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 \(\mathrm{m} .\)

A large 16.0 -kg roll of paper with radius \(R=18.0 \mathrm{cm}\) rests against the wall and is held in place by a bracket attached to a rod through the center of the roll (Fig. P10.69). The rod turns without friction in the bracket, and the moment of inertia of the paper and rod about the axis is 0.260 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) The other end of the bracket is attached by a frictionless hinge to the wall such that the bracket makes an angle of \(30.0^{\circ}\) with the wall. The weight of the bracket is negligible. The coefficient of kinetic friction between the paper and the wall is \(\mu_{\mathrm{k}}=0.25 .\) A constant vertical force \(F=60.0 \mathrm{N}\) is applied to the paper, and the paper unrolls. (a) What is the magnitude of the force that the rod exerts on the paper as it unrolls? (b) What is the magnitude of the angular acceleration of the roll?