Chapter 8

A wheel rolls uniformly on level ground without slipping. A piece of mud on the wheel flies off when it is at the 9 o'clock position (rear of wheel). Describe the subsequent motion of the mud.

A rope goes over a circular pulley with a radius of \(6.5 \mathrm{~cm}\). If the pulley makes 4 revolutions without the rope slipping, what length of rope passes over the pulley?

A wheel rolls 5 revolutions on a horizontal surface without slipping. If the center of the wheel moves \(3.2 \mathrm{~m}\), what is the radius of the wheel?

A bowling ball with a radius of \(15.0 \mathrm{~cm}\) travels down the lane so that its center of mass is moving at \(3.60 \mathrm{~m} / \mathrm{s} .\) The bowler estimates that it makes about 7.50 complete revolutions in 2.00 seconds. Is it rolling without slipping? Prove your answer, assuming that the bowler's quick observation limits answers to two significant figures.

A ball with a radius of \(15 \mathrm{~cm}\) rolls on a level surface, and the translational speed of the center of mass is \(0.25 \mathrm{~m} / \mathrm{s}\). What is the angular speed about the center of mass if the ball rolls without slipping?

(a) When a disk rolls without slipping, should the product \(r \omega\) be (1) greater than, (2) equal to, or (3) less than \(v_{\mathrm{CM}}\) ? (b) A disk with a radius of \(0.15 \mathrm{~m}\) rotates through \(270^{\circ}\) as it travels \(0.71 \mathrm{~m}\). Does the disk roll without slipping? Prove your answer.

A bocce ball with a diameter of \(6.00 \mathrm{~cm}\) rolls without slipping on a level lawn. It has an initial angular speed of \(2.35 \mathrm{rad} / \mathrm{s}\) and comes to rest after \(2.50 \mathrm{~m}\). Assuming constant deceleration, determine (a) the magnitude of its angular deceleration and (b) the magnitude of the maximum tangential acceleration of the ball's surface (tell where that part is located)

A cylinder with a diameter of \(20 \mathrm{~cm}\) rolls with an angular speed of \(0.050 \mathrm{rad} / \mathrm{s}\) on a level surface. If the cylinder experiences a uniform tangential acceleration of \(0.018 \mathrm{~m} / \mathrm{s}^{2}\) without slipping until its angular speed is \(1.2 \mathrm{rad} / \mathrm{s},\) through how many complete revolutions does the cylinder rotate during the time it accelerates?

The drain plug on a car's engine has been tightened to a torque of \(25 \mathrm{~m} \cdot \mathrm{N}\). If a 0.15 -m-long wrench is used to change the oil, what is the minimum force needed to loosen the plug?

How many different positions of stable equilibrium and unstable equilibrium are there for a cube? Consider each surface, edge, and corner to be a different position.

Two children are sitting on opposite ends of a uniform seesaw of negligible mass. (a) Can the seesaw be balanced if the masses of the children are different? How? (b) If a \(35-\mathrm{kg}\) child is \(2.0 \mathrm{~m}\) from the pivot point (or fulcrum), how far from the pivot point will her \(30-\mathrm{kg}\) playmate have to sit on the other side for the seesaw to be in equilibrium?

A uniform meterstick pivoted at its center, as in Example 8.5 , has a 100 -g mass suspended at the 25.0 -cm position. (a) At what position should a 75.0 - \(\mathrm{g}\) mass be suspended to put the system in equilibrium? (b) What mass would have to be suspended at the \(90.0-\mathrm{cm}\) position for the system to be in equilibrium?

Telephone and electrical lines are allowed to sag between poles so that the tension will not be too great when something hits or sits on the line. (a) Is it possible to have the lines perfectly horizontal? Why or why not? (b) Suppose that a line were stretched almost perfectly horizontally between two poles that are \(30 \mathrm{~m}\) apart. If a \(0.25-\mathrm{kg}\) bird perches on the wire midway between the poles and the wire sags \(1.0 \mathrm{~cm},\) what would be the tension in the wire? (Neglect the mass of the wire.)

A bowling ball (mass \(7.00 \mathrm{~kg}\) and radius \(17.0 \mathrm{~cm}\) ) is released so fast that it skids without rotating down the lane (at least for a while). Assume the ball skids to the right and the coefficient of sliding friction between the ball and the lane surface is \(0.400 .\) (a) What is the direction of the torque exerted by the friction on the ball about the center of mass of the ball? (b) Determine the magnitude of this torque (again about the ball's center of mass).

(a) How many uniform, identical textbooks of width \(25.0 \mathrm{~cm}\) can be stacked on top of each other on a level surface without the stack falling over if each successive book is displaced \(3.00 \mathrm{~cm}\) in width relative to the book below it? (b) If the books are \(5.00 \mathrm{~cm}\) thick, what will be the height of the center of mass of the stack above the level surface?

A \(10.0-\mathrm{kg}\) solid uniform cube with 0.500 -m sides rests on a level surface. What is the minimum amount of work necessary to put the cube into an unstable equilibrium position?

In a circus act, a uniform board (length \(3.00 \mathrm{~m}\), mass \(35.0 \mathrm{~kg}\) ) is suspended from a bungie-type rope at one end, and the other end rests on a concrete pillar. When a clown (mass \(75.0 \mathrm{~kg}\) ) steps out halfway onto the board, the board tilts so the rope end is \(30^{\circ}\) from the horizontal and the rope stays vertical. (a) In which situation will the rope tension be larger: (1) the board without the clown on it, (2) the board with the clown on it, or (3) you can't tell from the data given? (b) Calculate the force exerted by the rope in both situations.

A fixed 0.15-kg solid-disk pulley with a radius of \(0.075 \mathrm{~m}\) is acted on by a net torque of \(6.4 \mathrm{~m} \cdot \mathrm{N}\). What is the angular acceleration of the pulley?

What net torque is required to give a uniform \(20-\mathrm{kg}\) solid ball with a radius of \(0.20 \mathrm{~m}\) an angular acceleration of \(20 \mathrm{rad} / \mathrm{s}^{2} ?\)

A 2000 -kg Ferris wheel accelerates from rest to an angular speed of \(20 \mathrm{rad} / \mathrm{s}\) in \(12 \mathrm{~s}\). Approximate the Ferris wheel as a circular disk with a radius of \(30 \mathrm{~m}\). What is the net torque on the wheel?

Two objects of different masses are joined by a light rod. (a) Is the moment of inertia about the center of mass the minimum or the maximum? Why? (b) If the two masses are \(3.0 \mathrm{~kg}\) and \(5.0 \mathrm{~kg}\) and the length of the rod is \(2.0 \mathrm{~m},\) find the moments of inertia of the system about an axis perpendicular to the rod, through the center of the rod and the center of mass.

To start her lawn mower, Julie pulls on a cord that is wrapped around a pulley. The pulley has a moment of inertia about its central axis of \(I=0.550 \mathrm{~kg} \cdot \mathrm{m}^{2}\) and a radius of \(5.00 \mathrm{~cm}\). There is an equivalent frictional torque impeding her pull of \(\tau_{\mathrm{f}}=0.430 \mathrm{~m} \cdot \mathrm{N}\). To accelerate the pulley at \(\alpha=4.55 \mathrm{rad} / \mathrm{s}^{2},\) (a) how much torque does Julie need to apply to the pulley? (b) How much tension must the rope exert?

A meterstick pivoted about a horizontal axis through the 0 -cm end is held in a horizontal position and let go. (a) What is the initial tangential acceleration of the 100 cm position? Are you surprised by this result? (b) Which position has a tangential acceleration equal to the acceleration due to gravity?

A planetary space probe is in the shape of a cylinder. To protect it from heat on one side (from the Sun's rays), operators on the Earth put it into a "barbecue mode," that is, they set it rotating about its long axis. To do this, they fire four small rockets mounted tangentially as shown in \(v\) Fig. 8.51 (the probe is shown coming toward you). The object is to get the probe to rotate completely once every \(30 \mathrm{~s}\), starting from no rotation at all. They wish to do this by firing all four rockets for a certain length of time. Each rocket can exert a thrust of \(50.0 \mathrm{~N}\). Assume the probe is a uniform solid cylinder with a radius of \(2.50 \mathrm{~m}\) and a mass of \(1000 \mathrm{~kg}\) and neglect the mass of each rocket engine. Determine the amount of time the rockets need to be fired.

A constant retarding torque of \(12 \mathrm{~m} \cdot \mathrm{N}\) stops a rolling wheel of diameter \(0.80 \mathrm{~m}\) in a distance of \(15 \mathrm{~m}\). How much work is done by the torque?

A person opens a door by applying a 15-N force perpendicular to it at a distance \(0.90 \mathrm{~m}\) from the hinges. The door is pushed wide open (to \(120^{\circ}\) ) in \(2.0 \mathrm{~s}\). (a) How much work was done? (b) What was the average power delivered?

A constant torque of \(10 \mathrm{~m} \cdot \mathrm{N}\) is applied to the rim of a 10-kg uniform disk of radius \(0.20 \mathrm{~m}\). What is the angular speed of the disk about an axis through its center after it rotates 2.0 revolutions from rest?

A 2.5 -kg pulley of radius \(0.15 \mathrm{~m}\) is pivoted about an axis through its center. What constant torque is required for the pulley to reach an angular speed of \(25 \mathrm{rad} / \mathrm{s}\) after rotating 3.0 revolutions, starting from rest?

A solid ball of mass \(m\) rolls along a horizontal surface with a translational speed of \(v\). What percent of its total kinetic energy is translational?

You wish to accelerate a small merry-go-round from rest to a rotational speed of one-third of a revolution per second by pushing tangentially on it. Assume the merrygo-round is a disk with a mass of \(250 \mathrm{~kg}\) and a radius of \(1.50 \mathrm{~m} .\) Ignoring friction, how hard do you have to push tangentially to accomplish this in \(5.00 \mathrm{~s}\) ? (Use energy methods and assume a constant push on your part.)

A uniform sphere and a uniform cylinder with the same mass and radius roll at the same velocity side by side on a level surface without slipping. If the sphere and the cylinder approach an inclined plane and roll up it without slipping, will they be at the same height on the plane when they come to a stop? If not, what will be the percentage difference of the heights?

A hoop starts from rest at a height \(1.2 \mathrm{~m}\) above the base of an inclined plane and rolls down under the influence of gravity. What is the linear speed of the hoop's center of mass just as the hoop leaves the incline and rolls onto a horizontal surface? (Neglect friction.)

A cylindrical hoop, a cylinder, and a sphere of equal radius and mass are released at the same time from the top of an inclined plane. Using the conservation of mechanical energy, show that the sphere always gets to the bottom of the incline first with the fastest speed and that the hoop always arrives last with the slowest speed.

For the following objects, which all roll without slipping, determine the rotational kinetic energy about the center of mass as a percentage of the total kinetic energy: (a) a solid sphere, (b) a thin spherical shell, and (c) a thin cylindrical shell.

An industrial flywheel with a moment of inertia of \(4.25 \times 10^{2} \mathrm{~kg} \cdot \mathrm{m}^{2}\) rotates with a speed of \(7500 \mathrm{rpm}\) (a) How much work is required to bring the flywheel to rest? (b) If this work is done uniformly in \(1.5 \mathrm{~min}\), how much power is required?

A hollow, thin-shelled ball and a solid ball of equal mass are rolled up an inclined plane (without slipping) with both balls having the same initial velocity at the bottom of the plane. (a) Which ball rolls higher on the incline before coming to rest? (b) Do the radii of the balls make a difference? (c) After stopping, the balls roll back down the incline. By the conservation of energy, both balls should have the same speed when reaching the bottom of the incline. Show this explicitly.

In a tumbling clothes dryer, the cylindrical drum (radius \(50.0 \mathrm{~cm}\) and mass \(35.0 \mathrm{~kg}\) ) rotates once every second. (a) Determine the rotational kinetic energy about its central axis. (b) If it started from rest and reached that speed in \(2.50 \mathrm{~s}\), determine the average net torque on the dryer drum.

A steel ball rolls down an incline into a loop-theloop of radius \(R\) (v Fig. 8.52a). (a) What minimum speed must the ball have at the top of the loop in order to stay on the track? (b) At what vertical height \((h)\) on the incline, in terms of the radius of the loop, must the ball be released in order for it to have the required minimum speed at the top of the loop? (Neglect frictional losses.) (c) Figure \(8.52 \mathrm{~b}\) shows the loop-the-loop of a roller coaster. What are the sensations of the riders if the roller coaster has the minimum speed or a greater speed at the top of the loop? [Hint: In case the speed is below the minimum, seat and shoulder straps hold the riders in.

A 10 -kg rotating disk of radius \(0.25 \mathrm{~m}\) has an angular momentum of \(0.45 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}\) What is the angular speed of the disk?

Circular disks are used in automobile clutches and transmissions. When a rotating disk couples to a stationary one through frictional force, the energy from the rotating disk can transfer to the stationary one. (a) Is the angular speed of the coupled disks (1) greater than, (2) less than, or (3) the same as the angular speed of the original rotating disk? Why? (b) If a disk rotating at 800 rpm couples to a stationary disk with three times the moment of inertia, what is the angular speed of the combination?

An ice skater has a moment of inertia of \(100 \mathrm{~kg} \cdot \mathrm{m}^{2}\) when his arms are outstretched and a moment of inertia of \(75 \mathrm{~kg} \cdot \mathrm{m}^{2}\) when his arms are tucked in close to his chest. If he starts to spin at an angular speed of 2.0 rps (revolutions per second) with his arms outstretched, what will his angular speed be when they are tucked in?

An ice skater spinning with outstretched arms has an angular speed of \(4.0 \mathrm{rad} / \mathrm{s}\). She tucks in her arms, decreasing her moment of inertia by \(7.5 \% .\) (a) What is the resulting angular speed? (b) By what factor does the skater's kinetic energy change? (Neglect any frictional effects.) (c) Where does the extra kinetic energy come from?

While repairing his bicycle, a student turns it upside down and sets the front wheel spinning at 2.00 rev \(/ \mathrm{s}\). Assume the wheel has a mass of \(3.25 \mathrm{~kg}\) and all of the mass is located on the rim, which has a radius of \(41.0 \mathrm{~cm}\). To slow the wheel, he places his hand on the tire, thereby exerting a tangential force of friction on the wheel. It takes \(3.50 \mathrm{~s}\) to come to rest. Use the change in angular momentum to determine the force he exerts on the wheel. Assume the frictional force of the axle is negligible.

A kitten stands on the edge of a lazy Susan (a turntable). Assume that the lazy Susan has frictionless bearings and is initially at rest. (a) If the kitten starts to walk around the edge of the lazy Susan, the lazy Susan will (1) remain lazy and stationary, (2) rotate in the direction opposite that in which the kitten is walking, or (3) rotate in the direction the kitten is walking. Explain. (b) The mass of the kitten is \(0.50 \mathrm{~kg},\) and the lazy Susan has a mass of \(1.5 \mathrm{~kg}\) and a radius of \(0.30 \mathrm{~m}\). If the kitten walks at a speed of \(0.25 \mathrm{~m} / \mathrm{s},\) relative to the ground, what will be the angular speed of the lazy Susan? (c) When the kitten has walked completely around the edge and is back at its starting point, will that point be above the same point on the ground as it was at the start? If not, where is the kitten relative to the starting point? (Speculate on what might happen if everyone on the Earth suddenly started to run eastward. What effect might this have on the length of a day?)

IE A small heavy object of mass \(m\) is attached to a thin string to make a simple pendulum whose length is \(L\) When the object is pulled aside by a horizontal force \(F\) it is in static equilibrium and the string makes a constant angle \(\theta\) from the vertical. (a) The tension in the string should be (1) the same as, (2) greater than, or (3) less than the object's weight, \(m g .\) (b) Use the force condition for static equilibrium (along with a free-body diagram of the object) to prove that the string tension is \(T=\frac{m g}{\cos \theta}>m g\). Use the same procedure to show that \(F=m g \tan \theta\). (c) Prove the same result for \(F\) as in part (b) using the torque condition, summing the torques about the string's tied end. Explain why you cannot use this method to determine the string tension.

A bowling ball with a diameter of \(21.6 \mathrm{~cm}\) is rolling down a level alley surface at \(12.7 \mathrm{~m} / \mathrm{s}\) without slipping. Assume the ball is uniform and made of plastic with a density of \(800 \mathrm{~kg} / \mathrm{m}^{3}\). (a) What is the angular speed of the ball? (b) Calculate the speed (relative to the alley surface) of a point on top of the ball directly above the contact point on the floor. (c) What is the ball's linear kinetic energy? (d) If it now starts to roll up a \(30^{\circ}\) incline, how far up the incline will it travel before it stops?

A flat cylindrical grinding wheel is spinning at 2000 rpm (clockwise when viewed head-on) when its power is suddenly turned off. Normally, if left alone, it takes 45.0 s to coast to rest. Assume the grinder has a moment of inertia of \(2.43 \mathrm{~kg} \cdot \mathrm{m}^{2}\). (a) Determine its angular acceleration during this process. (b) Determine the tangential acceleration of a point on the grinding wheel if the wheel is \(7.5 \mathrm{~cm}\) in diameter. (c) The slowing down is caused by a frictional torque on the axle of the wheel. The axle is \(1.00 \mathrm{~cm}\) in diameter. Determine the frictional force on the axle. (d) How much work was done by friction on the system?

A flat, solid cylindrical grinding wheel with a diameter of \(20.2 \mathrm{~cm}\) is spinning at 3000 rpm when its power is suddenly turned off. A workman continues to press his tool bit toward the wheel's center at the wheel's circumference so as to continue to grind as the wheel coasts to a stop. If the wheel has a moment of inertia of \(4.73 \mathrm{~kg} \cdot \mathrm{m}^{2}\), (a) determine the necessary torque that must be exerted by the workman to bring it to rest in \(10.5 \mathrm{~s}\). Ignore any friction at the axle. (b) If the coefficient of kinetic friction between the tool bit and the wheel surface is \(0.85,\) how hard must the workman push on the bit?

A uniform sphere of mass \(2.50 \mathrm{~kg}\) and radius \(15.0 \mathrm{~cm}\) is released from rest at the top of an incline that is \(5.25 \mathrm{~m}\) long and makes an angle of \(35^{\circ}\) with the horizontal. Assuming it rolls without slipping, (a) determine its total kinetic energy at the bottom of the incline. (b) Determine its rotational kinetic energy at the bottom of the incline. (c) What type of friction, static or kinetic, is acting on the surface of the sphere? Explain. (d) Determine the force of friction in part (d).

A stationary ice skater with a mass of \(80.0 \mathrm{~kg}\) and a moment of inertia (about her central vertical axis) of \(3.00 \mathrm{~kg} \cdot \mathrm{m}^{2}\) catches a baseball with her outstretched arm. The catch is made at a distance of \(1.00 \mathrm{~m}\) from the central axis. The ball has a mass of \(145 \mathrm{~g}\) and is traveling at \(20.0 \mathrm{~m} / \mathrm{s}\) before the catch. (a) What linear speed does the system (skater \(+\) ball) have after the catch? (b) What is the angular speed of the system (skater \(+\) ball) after the catch? (c) What percentage of the ball's initial kinetic energy is lost during the catch? Neglect friction with the ice.