Chapter 10

A block with mass \(m=5.00 \mathrm{kg}\) slides down a a surface inclined \(36.9^{\circ}\) to the horizontal (Fig. P10.70). The coefficient of kinetic friction is \(0.25 .\) A string attached to the block is wrapped around a flywheel on a fixed axis at \(O .\) The flywheel has mass 25.0 \(\mathrm{kg}\) and moment of inertia 0.500 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) with respect to the axis of rotation. The string pulls without slipping at a perpendicular distance of 0.200 \(\mathrm{m}\) from that axis. (a) What is the acceleration of the block down the plane? (b) What is the tension in the string?

A lawn roller in the form of a thin-walled, hollow cylinder with mass \(M\) is pulled horizontally with a constant horizontal force \(F\) applied by a handle attached to the axle. If it rolls without slipping, find the acceleration and the friction force.

A solid disk is rolling without slipping on a level surface at a constant speed of 3.60 \(\mathrm{m} / \mathrm{s}\) (a) If the disk rolls up a \(30.0^{\circ}\) ramp, how far along the ramp will it move before it stops? (b) Explain why your answer in part (a) does not depend on either the mass or the radius of the disk.

The Yo-yo. A yo-yo is made from two uniform disks, each with mass \(m\) and radius \(R\) , connected by a light axle of radius \(b .\) A light, thin string is wound several times around the axle and then held stationary while the yo-yo is released from rest, dropping as the string unwinds. Find the linear acceleration and angular acceleration of the yo-yo and the tension in the string.

Rolling Stones. A solid, uniform, spherical boulder starts from rest and rolls down a \(50.0-\mathrm{m}\) -high hill, as shown in Fig. Plo.81. The top half of the hill is rough enough to cause the boulder to roll without slipping, but the lower half is covered with ice and there is no friction. What is the translational speed of the boulder when it reaches the bottom of the hill?

A 42.0 -cm-diameter wheel, consisting of a rim and six spokes, is constructed from a thin, rigid plastic material having a linear mass density of 25.0 \(\mathrm{g} / \mathrm{cm} .\) This wheel is released from rest at the top of a hill 58.0 \(\mathrm{m}\) high. (a) How fast is it rolling when it reaches the bottom of the hill? (b) How would your answer change if the linear mass density and the diameter of the wheel were each doubled?

A child rolls a 0.600 -kg basketball up a long ramp. The basketball can be considered a thin-walled, hollow sphere. When the child releases the basketball at the bottom of the ramp, it has a speed of 8.0 \(\mathrm{m} / \mathrm{s} .\) When the ball returns to her after rolling up the ramp and then rolling back down, it has a speed of 4.0 \(\mathrm{m} / \mathrm{s}\) . Assume the work done by friction on the basketball is the same when the ball moves up or down the ramp and that the basketball rolls without slipping. Find the maximum vertical height increase of the ball as it rolls up the ramp.

In a lab experiment you let a uniform ball roll down a curved track. The ball starts from rest and rolls without slipping.While on the track, the ball descends a vertical distance \(h .\) The lower end of the track is horizontal and extends over the edge of the lab table; the ball leaves the track traveling horizontally. While free-falling after leaving the track, the ball moves a horizontal distance \(x\) and a vertical distance \(y\) . (a) Calculate \(x\) in terms of \(h\) and \(y\) ignoring the work done by friction. (b) Would the answer to part (a) be any different on the moon? (c) Although you do the experiment very carefully, your measured value of \(x\) is consistently a bit smaller than the value calculated in part (a). Why? (d) What would \(x\) be for the same \(h\) and \(y\) as in part (a) if you let a silver dollar roll down the track? You can ignore the work done by friction.

A uniform solid cylinder with mass \(M\) and radius 2\(R\) rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley with mass \(M\) and radius \(R\) that is mounted on a frictionless axle through its center. A block of mass \(M\) is suspended from the free end of the string (Fig. P10.87). The string doesn't slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest.

A uniform, \(0.0300-\mathrm{kg}\) rod of length 0.400 \(\mathrm{m}\) rotates in a horizontal plane about a fixed axis through its center and perpendicular to the rod. Two small rings, each with mass 0.0200 \(\mathrm{kg}\) , are mounted so that they can slide along the rod. They are initially held by catches at positions 0.0500 \(\mathrm{m}\) on each side of the center of the rod, and the system is rotating at 30.0 rev/min. With no other changes in the system, the catches are released, and the rings slide outward along the rod and fly off at the ends. (a) What is the angular speed of the system at the instant when the rings reach the ends of the rod? (b) What is the angular speed of the rod after the rings leave it?

Tarzan and Jane in the 21 st Century. Tarzan has foolishly gotten himself into another scrape with the animals and must be rescued once again by Jane. The 60.0 -kg Jane starts from rest at a height of 5.00 \(\mathrm{m}\) in the trees and swings down to the ground using a thin, but very rigid, 30.0 -kg vine 8.00 \(\mathrm{m}\) long. She arrives just in time to snatch the 72.0 -kg Tarzan from the jaws of an angry hippopotamus. What is Jane's (and the vine's) angular speed (a) just before she grabs Tarzan and (b) just after she grabs him? (c) How high will Tarzan and Jane go on their first swing after this daring rescue?

A uniform rod of length \(L\) rests on a frictionless horizontal surface. The rod pivots about a fixed frictionless axis at one end. The rod is initially at rest. A bullet traveling parallel to the horizontal surface and perpendicular to the rod with speed \(v\) strikes the rod at its center and becomes embedded in it. The mass of the bullet is one-fourth the mass of the rod. (a) What is the final angeed of the rod? (b) What is the ratio of the kinetic energy of the system after the collision to the kinetic energy of the bullet before the collision?

The solid wood door of a gymnasium is 1.00 \(\mathrm{m}\) wide and 2.00 \(\mathrm{m}\) high, has total mass \(35.0 \mathrm{kg},\) and is hinged along one side. The door is open and at rest when a stray basketball hits the center of the door head-on, applying an average force of 1500 \(\mathrm{N}\) to the door for 8.00 \(\mathrm{ms} .\) Find the angular speed of the door after the impact. [Hint: Integrating Eq. \((10.29)\) yields \(\Delta L_{z}=\int_{t_{1}}^{t_{2}}\left(\Sigma \tau_{z}\right) d t=\left(\sum \tau_{z}\right)_{\mathrm{av}} \Delta t .\) The quantity \(\int_{t_{1}}^{t_{2}}\left(\Sigma \tau_{z}\right) d t\) is called the angular impulse.]

A target in a shooting gallery consists of a vertical square wooden board, 0.250 \(\mathrm{m}\) on a side and with mass 0.750 \(\mathrm{kg}\) , that pivots on a horizontal axis along its top edge. The board is struck face-on at its center by a bullet with mass 1.90 \(\mathrm{g}\) that is traveling at 360 \(\mathrm{m} / \mathrm{s}\) and that remains embedded in the board. (a) What is the angular speed of the board just after the bullet's impact? (b) What maximum height above the equilibrium position does the center of the board reach before starting to swing down again? (c) What minimum bullet speed would be required for the board to swing all the way over after impact?

Neutron Star Glitches. Occasionally, a rotating neutron star (see Exercise 10.41 ) undergoes a sudden and unexpected speedup called a glitch. One explanation is that a glitch occurs when the crust of the neutron star settles slightly, decreasing the moment of inertia about the rotation axis. A neutron star with angular speed \(\omega_{0}=70.4 \mathrm{rad} / \mathrm{s}\) underwent such a glitch in October 1975 that increased its angular speed to \(\omega=\omega_{0}+\Delta \omega,\) where \(\Delta \omega / \omega_{0}=2.01 \times 10^{-6} .\) If the radius of the neutron star before the glitch was \(11 \mathrm{km},\) by how much did its radius decrease in the star- quake? Assume that the neutron star is a uniform sphere.

A \(500.0-\mathrm{g}\) bird is flying horizontally at 2.25 \(\mathrm{m} / \mathrm{s}\) not paying much attention, when it suddenly flies into a stationary vertical bar, hitting it 25.0 \(\mathrm{cm}\) below the top (Fig. P10.95). The bar is uniform, 0.750 \(\mathrm{m}\) long, has a mass of \(1.50 \mathrm{kg},\) and is hinged at its base. The collision stuns the bird so that it just drops to the ground afterward (but soon recovers to fly happily away). What is the angular velocity of the bar (a) just after it is hit by the bird and (b) just as it reaches the ground?

A small block with mass 0.250 \(\mathrm{kg}\) is attached to a string passing through a hole in a frictionless, horizontal surface (see Fig. E10.42). The block is originally revolving in a circle with a radius of 0.800 m about the hole with a tangential speed of 4.00 \(\mathrm{m} / \mathrm{s}\) . The string is then pulled slowly from below, shortening the radius of the circle in which the block revolves. The breaking strength of the string is 30.0 \(\mathrm{N} .\) What is the radius of the circle when the string breaks?

A horizontal plywood disk with mass 7.00 \(\mathrm{kg}\) and diameter 1.00 \(\mathrm{m}\) pivots on frictionless bearings about a vertical axis through its center. You attach a circular model-railroad track of negligible mass and average diameter 0.95 m to the disk. A 1.20 -kg, battery-driven model train rests on the tracks. To demonstrate conservation of angular momentum, you switch on the train's engine. The train moves counterclockwise, soon attaining a constant speed of 0.600 \(\mathrm{m} / \mathrm{s}\) relative to the tracks. Find the magnitude and direction of the angular velocity of the disk relative to the earth.

A 55-kg runner runs around the edge of a horizontal turntable mounted on a vertical, frictionless axis through its center. The runner's velocity relative to the earth has magnitude 2.8 \(\mathrm{m} / \mathrm{s}\) . The turntable is rotating in the opposite direction with an angular velocity of magnitude 0.20 rad/s relative to the earth. The radius of the turntable is \(3.0 \mathrm{m},\) and its moment of inertia about the axis of rotation is 80 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) Find the final angular velocity of the system if the runner comes to rest relative to the turntable. (You can model the runner as a particle.)

A uniform ball of radius \(R\) rolls without slipping between two rails such that the horizontal distance is \(d\) between the two contact points of the rails to the ball. (a) In a sketch, show that at any instant \(v_{\mathrm{cm}}=\omega \sqrt{R^{2}-d^{2} / 4} .\) Discuss this expression in the limits \(d=0\) and \(d=2 R\) . (b) For a uniform ball starting from rest and descending a vertical distance \(h\) while rolling without slipping down a ramp, \(v_{\mathrm{cm}}=\sqrt{10 g h / 7}\) . Replacing the ramp with the two rails, show that $$ v_{\mathrm{cm}}=\sqrt{\frac{10 g h}{5+2 /\left(1-d^{2} / 4 R^{2}\right)}} $$ In each case, the work done by friction has been ignored. (c) Which speed in part (b) is smaller? Why? Answer in terms of how the loss of potential energy is shared between the gain in translational and rotational kinetic energies. (d) For which value of the ratio \(d / R\) do the two expressions for the speed in part (b) differ by 5.0\(\% ?\) By 0.50\(\% ?\)

When an object is rolling without slipping, the rolling friction force is much less the friction force when the object is sliding; a silver dollar will roll on its edge much farther than it will slide on its flat side (see Section 5.3\() .\) When an object is rolling without slipping on a horizontal surface, we can approximate the friction force to be zero, so that \(a_{x}\) and \(\alpha_{z}\) are approximately zero and \(v_{x}\) and \(\omega_{z}\) are approximately constant. Rolling without slipping means \(v_{x}=r \omega_{z}\) and \(a_{x}=r \alpha_{z}\) . If an object is set in motion on a surface without these equalities, sliding (kinetic) friction will act on the object as it slips until rolling without slipping is established. A solid cylinder with mass \(M\) and radius \(R\) , rotating with angular speed \(\omega_{0}\) about an axis through its center, is set on a horizontal speed \(\omega_{0}\) about the kinetic friction coefficient is \(\mu_{\mathrm{k}}\) (a) Draw a free-body diagram for the cylinder on the surface. Think carefully about the direction of the kinetic friction force. on the cylinder. Calculate the accelerations \(a_{x}\) of the center of mass and \(\alpha_{z}\) of rotation about the center of mass. (b) The cylinder is initially slipping completely, so initially \(\omega_{z}=\omega_{0}\) but \(v_{x}=0 .\) Rolling without slipping sets in when \(v_{x}=R \omega_{z} .\) Calculate the distance the cylinder rolls before slipping stops. (c) Calculate the work done by the friction force on the cylinder as it moves from where it was set down to where it begins to roll without slipping.

A demonstration gyroscope wheel is constructed by removing the tire from a bicycle wheel 0.650 \(\mathrm{m}\) in diameter, wrapping lead wire around the rim, and taping it in place. The shaft projects 0.200 \(\mathrm{m}\) at each side of the wheel, and a woman holds the ends of the shaft in her hands. The mass of the system is 8.00 \(\mathrm{kg}\) ; its entire mass may be assumed to be located at its rim. The shaft is horizontal, and the wheel is spinning about the shaft at 5.00 \(\mathrm{rev} / \mathrm{s}\) . Find the magnitude and direction of the force each hand exerts on the shaft (a) when the shaft is at rest; (b) when the shaft is rotating in a horizontal plane about its center at \(0.050 \mathrm{rev} / \mathrm{s} ;\) (c) when the shaft is rotating in a horizontal plane about its center at 0.300 \(\mathrm{rev} / \mathrm{s}\) . (d) At what rate must the shaft rotate in order that it may be supported at one end only?