Solved: The moment of inertia of a uniform rod about an

Problem 1CQ Figure Q7.3 shows four pulleys, each with a heavy and a light block strung over it. The blocks’ velocities are shown. What are the signs (+ or -) of the angular velocity and angular acceleration of the pulley in each case?

Problem 1P To throw a discus, the thrower holds it with a fully outstretched arm. Starting from rest, he begins to turn with a constant angular acceleration, releasing the discus after making one complete revolution. The diameter of the circle in which the discus moves is about 1.8 m. If the thrower takes 1.0 s to complete one revolution, starting from rest, what will be the speed of the discus at release?

Problem 2CQ If you are using a wrench to loosen a very stubborn nut, you can make the job easier by using a “cheater pipe.” This is a piece of pipe that slides over the handle of the wrench, as shown in Figure Q7.4 , making it effectively much longer. Explain why this would help you loosen the nut.

Problem 2P A computer hard disk starts from rest, then speeds up with an angular acceleration of until it reaches its final angular speed of 7200 rpm. How many revolutions has the disk made 10.0 s after it starts up?

Problem 3CQ Five forces are applied to a door, as seen from above in Figure Q7.5 . For each force, is the torque about the hinge positive, negative, or zero?

Problem 3P The crankshaft in a race car goes from rest to 3000 rpm in 2.0 s. a. What is the crankshaft’s angular acceleration? b. How many revolutions does it make while reaching 3000 rpm?

Problem 4CQ A screwdriver with a very thick handle requires less force to operate than one with a very skinny handle. Explain why this is so.

Problem 4P Reconsider the situation in Example 7.10. If Luis pulls straight down on the end of a wrench that is in the same orientation but is 35 cm long, rather than 20 cm, what force must he apply to exert the same torque?

Problem 5CQ If you have ever driven a truck, you likely found that it had a steering wheel with a larger diameter than that of a passenger car. Why is this?

Problem 5P Balls are attached to light rods and can move in horizontal circles as shown in Figure P7.17. Rank in order, from smallest to largest, the torques about the centers of the circles. Explain

Problem 6CQ A common type of door stop is a wedge made of rubber. Is such a stop more effective when jammed under the door near or far from the hinges? Why?

Problem 7CQ Suppose you are hanging from a tree branch. If you move out along the branch, farther away from the trunk, the branch will be more likely to break. Explain why this is so.

Problem 7P What is the net torque about the axle on the pulley in Figure P7.19?

Problem 8CQ A student gives a quick push to a ball at the end of a mass-less, rigid rod, causing the ball to rotate clockwise in a horizontal circle as shown in Figure 8. The rod's pivot is frictionless. a. As the student is pushing, is the torque about the pivot positive, negative, or zero? b. After the push has ended, what does the ball's angular velocity do? Steadily increase? Increase for a while, then hold steady? Hold steady? Decrease for a while, then hold steady? Steadily decrease? Explain. c. Right alter the push has ended, is the torque positive, negative, or zero? FIGURE 8

Problem 6P Six forces, each of magnitude either F or 2F, are applied to a door as seen from above in Figure P7.18. Rank in order, from smallest to largest, the six torques about the hinge.

Problem 8P The tune-up specifications of a car call for the spark plugs to be tightened to a torque of 38 N . m. You plan to tighten the plugs by pulling on the end of a 25-cm-long wrench. Because of the cramped space under the hood, you’ll need to pull at an angle of 120° with respect to the wrench shaft. With what force must you pull?

Problem 9CQ The two ends of the dumbbell shown in Figure Q7.11 are made of the same material. Is the dumbbell’s center of gravity at point 1, 2, or 3? Explain.

Problem 9P A professor’s office door is 0.91 m wide, 2.0 m high, and 4.0 cm thick; has a mass of 25 kg; and pivots on frictionless hinges. A “door closer” is attached to the door and the top of the door frame. When the door is open and at rest, the door closer exerts a torque of 5.2 N . m. What is the least force that you need to apply to the door to hold it open?

Problem 10CQ When you rise from a chair, you have to lean quite far forward (try it!). Why is this?

Problem 10P In Figure P7.22, force acts half as far from the pivot as . What magnitude of causes the net torque on the rod to be zero?

Problem 10P In Figure P7.22, force acts half as far from the pivot as . What magnitude of causes the net torque on the rod to be zero?

Problem 11P Tom and Jerry both push on the 3.00-m-diameter merry-go-round shown in Figure P7. 11. a. If Tom pushes with a force of 40.0 N and Jerry pushes with a force of 35.2 N, what is the net torque on the merry-go-round? b. What is the net torque if Jerry reverses the direction he pushes by 180° without changing the magnitude of his force? FIGURE 11

Problem 12CQ The moment of inertia of a uniform rod about an axis through its center is . The moment of inertia about an axis at one end is . Explain why the moment of inertia is larger about the end than about the center.

Problem 12P What is the net torque on the bar shown in Figure P7.24, about the axis indicated by the dot?

Problem 13CQ A heavy steel rod, 1.0 m long, and a light pencil, 0.15 m long, are held 15° from the vertical with one end on a table, then released simultaneously. Which will hit the table first? Or will it be a tie? Explain.

Problem 13P What is the net torque on the bar shown in Figure P7.25, about the axis indicated by the dot?

The wheel in Figure Q7.14 is rolling to the right without sl ipping. Rank in order, from fastest to slowest, the speeds of the points labeled I through 5. Ex plain your reasoning.

Problem 14P What is the net torque on the bar shown in Figure P7.26, about the axis indicated by the dot?

Problem 15CQ A car traveling at 60 mph has a pebble stuck in one of its tires. Eventually the pebble works loose, and at the instant of release it is at the top of the tire. Explain why the pebble then slams hard into the front of the wheel well.

Problem 15P A 1.7-m-long barbell has a 20 kg weight on its right end and a 35 kg weight on its left end. a. If you ignore the weight of the bar itself, how far from the left end of the barbell is the center of gravity? b. Where is the center of gravity if the 8.0 kg mass of the barbell itself is taken into account?

Problem 16MCQ A nut needs to be tightened with a wrench. Which force shown in Figure Q7.17 will apply the greatest torque to the nut?

Three identical coins lie on three corners of a square 10.0 cm on a side, as shown in Figure P7.28. Determine the - and -coordinates of the center of gravity of the group of three coins.

Problem 17MCQ Suppose a bolt on your car engine needs to be tightened to a torque of 20 N. m. You are using a 15-cm-long wrench, and you apply a force at the very end in the direction that produces maximum torque. What force should you apply? A. 1300 N B. 260 N C. 130 N D. 26 N

Problem 17P Hold your arm outstretched so that it is horizontal. Estimate the mass of your arm and the position of its center of gravity. What is the gravitational torque on your arm in this position, computed around the shoulder joint?

Problem 18MCQ A machine part is made up of two pieces, with centers of gravity shown in Figure Q7.19 . Which point could be the center of gravity of the entire part?

Problem 18P A solid cylinder sits on top of a solid cube as shown in Figure P7.30. How far above the table’s surface is the center of gravity of the combined object?

Problem 19MCQ A typical compact disk has a mass of 15 g and a diameter of 120 mm. What is its moment of inertia about an axis through its center, perpendicular to the disk?

Problem 19P The 2.0 kg, uniform, horizontal rod in Figure P7.31 is seen from the side. What is the gravitational torque about the point shown?

Problem 20MCQ Suppose you make a new kind of compact disk that is the same thickness as a current disk but twice the diameter. By what factor will the moment of inertia increase? A. 2 B. 4 C. 8 D. 16

Problem 20P A 4.00-m-long, 500 kg steel beam extends horizontally from the point where it has been bolted to the framework of a new building under construction. A 70.0 kg construction worker stands at the far end of the beam. What is the magnitude of the gravitational torque about the point where the beam is bolted into place?

Problem 21MCQ Doors 1 and 2 have the same mass, height, and thickness. Door 2 is twice as wide as door 1. Bob pushes straight against the outer edge of door 2 with force F, and Barb pushes straight against the outer edge of door 1 with force 2F. How do the angular accelerations ?1 and ?2 of the two doors compare? A. ?1 > ?2 B. ?1 = ?2 C. ?1 < ?1

Problem 21P An athlete at the gym holds a 3.0 kg steel ball in his hand. His arm is 70 cm long and has a mass of 4.0 kg. What is the magnitude of the gravitational torque about his shoulder if he holds his arm a. Straight out to his side, parallel to the floor? b. Straight, but 45° below horizontal?

Problem 22MCQ A baseball bat has a thick, heavy barrel and a thin, light handle. If you want to hold a baseball bat on your palm so that it balances vertically, you should A. Put the end of the handle in your palm, with the barrel up. B. Put the end of the barrel in your palm, with the handle up. C. The bat will be equally easy to balance in either configuration.

The \(2.0\)-m-long, \(15 \mathrm{~kg}\) beam in Figure P7.34 is hinged at its left end. It is "falling" (rotating clockwise, under the influence of gravity), and the figure shows its position at three different times. What is the gravitational torque on the beam about an axis through the hinged end when the beam is at the a. Upper position? b. Middle position? c. Lower position?

Problem 23MCQ A car traveling at a steady 30 m/s has 74-cm-diameter tires. What is the approximate acceleration of a piece of the tread on any of the tires? A. 24 m/s2 B. 48 m/s2 C. 2400 m/s2 D. 4800 m/s2

Problem 23P Two thin beams are joined end-to-end as shown in Figure P7.35 to make a single object. The left beam is 10.0 kg and 1.00 m long and the right one is 40.0 kg and 2.00 m long. a. How far from the left end of the left beam is the center of gravity of the object? b. What is the gravitational torque on the object about an axis through its left end? The object is seen from the side.

Problem 24P Figure P7.36 shows two thin beams joined at right angles. The vertical beam is 15.0 kg and 1.00 m long and the horizontal beam is 25.0 kg and 2.00 m long. a. Find the center of gravity of the two joined beams. Express your answer in the form (x, y), taking the origin at the corner where the beams join. b. Calculate the gravitational torque on the joined beams about an axis through the corner. The beams are seen from the side.

Problem 25P A regulation table tennis ball is a thin spherical shell 40 mm in diameter with a mass of 2.7 g. What is its moment of inertia about an axis that passes through its center?

Problem 26P Three pairs of balls are connected by very light rods as shown in Figure P7.38. Rank in order, from smallest to largest, the moments of inertia about axes through the centers of the rods.

Problem 27P A playground toy has four seats, each 5.0 kg, attached to very light, 1.5-m-long rods, as seen from above in Figure P7.39. If two children, with masses of 15 kg and 20 kg, sit in seats opposite one another, what is the moment of inertia about the rotation axis?

Problem 28P A solid cylinder with a radius of 4.0 cm has the same mass as a solid sphere of radius R. If the cylinder and sphere have the same moment of inertia about their centers, what is the sphere’s radius?

Problem 29P A bicycle rim has a diameter of 0.65 m and a moment of inertia, measured about its center, of . What is the mass of the rim?

Problem 30P The left part of Figure 30 shows a bird's-eye view of two identical balls connected by a light rod that rotates about a vertical axis through its center. The right part shows a ball of twice the mass connected to a light rod of half the length, that rotates about its left end. If equal forces are applied as shown in the figure, which of the two rods will have the greater angular acceleration? FIGURE 30

Problem 31P a. What is the moment of inertia of the door in Problem 21? b. If you let go of the open door, what is its angular acceleration immediately afterward? Reference: Problem 21: A professor’s office door is 0.91 m wide, 2.0 m high, and 4.0 cm thick; has a mass of 25 kg; and pivots on frictionless hinges. A “door closer” is attached to the door and the top of the door frame. When the door is open and at rest, the door closer exerts a torque of 5.2 N . m. What is the least force that you need to apply to the door to hold it open?

Problem 32P A small grinding wheel has a moment of inertia . What net torque must be applied to the wheel for its angular acceleration to be ?

Problem 33P While sitting in a swivel chair, you push against the floor with your heel to make the chair spin. The 7.0 N frictional force is applied at a point 40 cm from the chair’s rotation axis, in the direction that causes the greatest angular acceleration. If that angular acceleration is , what is the total moment of inertia about the axis of you and the chair?

Problem 34P An object’s moment of inertia is . Its angular velocity is increasing at the rate of 4.0 rad/s per second. What is the net torque on the object?

Problem 35P A 200 g, 20-cm-diameter plastic disk is spun on an axle through its center by an electric motor. What torque must the motor supply to take the disk from 0 to 1800 rpm in 4.0 s?

Problem 36P The 2.5 kg object shown in Figure P7.47 has a moment of inertia about the rotation axis of . The rotation axis is horizontal. When released, what will be the object’s initial angular acceleration?

Problem 37P A frictionless pulley, which can be modeled as a 0.80 kg solid cylinder with a 0.30 m radius, has a rope going over it, as shown in Figure P7.48. The tension in the rope is 10 N on one side and 12 N on the other. What is the angular acceleration of the pulley?

Problem 38P If you lift the front wheel of a poorly maintained bicycle off the ground and then start it spinning at 0.72 rev/s, friction in the bearings causes the wheel to stop in just 12 s. If the moment of inertia of the wheel about its axle is , what is the magnitude of the frictional torque?

Problem 39P A toy top with a spool of diameter 5.0 cm has a moment of inertia of about its rotation axis. To get the top spinning, its string is pulled with a tension of 0.30 N. How long does it take for the top to complete the first five revolutions? The string is long enough that it is wrapped around the top more than five turns.

Problem 40P A 34-cm-diameter potter's wheel with a mass of 20 kg is spinning at 180 rpm. Using her hands, a potter forms a pot, centered on the wheel, with a 14 cm diameter. Her hands apply a net friction force of 1.3 N to the edge of the pot. If the power goes out, so that the wheel's motor no longer provides any torque, how long will it take for the wheel to come to a stop in her hands?

Problem 41P A 1.5 kg block and a 2.5 kg block are attached to opposite ends of a light rope. The rope hangs over a solid, frictionless pulley that is 30 cm in diameter and has a mass of 0.75 kg. When the blocks are released, what is the acceleration of the lighter block?

Problem 42P A bicycle with 0.80-m-diameter tires is coasting on a level road at 5.6 m/s. A small blue dot has been painted on the tread of the rear tire. a. What is the angular speed of the tires? b. What is the speed of the blue dot when it is 0.80 m above the road? c. What is the speed of the blue dot when it is 0.40 m above the road?

Problem 43P A 1.2 g pebble is stuck in a tread of a 0.76-m-diameter automobile tire, held in place by static friction mat can be at most 3.6 N. The car starts from rest and gradually accelerates on a straight road. How fast is the car moving when the pebble flies out of the tire tread?

Problem 44GP Figure P7.55 shows the angular position-versus-time graph for a particle moving in a circle. a. Write a description of the particle’s motion. b. Draw the angular velocity-versus-time graph.

Problem 45GP The graph in Figure P7.56 shows the angular velocity of the crankshaft in a car. Draw a graph of the angular acceleration versus time. Include appropriate numerical scales on both axes.

Problem 46GP Problem A computer disk is 8.0 cm in diameter. A reference dot on the edge of the disk is initially located at ? = 45°. The disk accelerates steadily for 0.50 s, reaching 2000 rpm, then coasts at steady angular velocity for another 0.50 s. a. What is the tangential acceleration of the reference dot at t = 0.25 s? ________________ b. What is the centripetal acceleration of the reference dot at t = 0.25 s? ________________ c. What i s the angular position of the reference dot at t ? 1.0 s? ________________ d. What is the speed of the reference dot at t = 1.0 s?

Problem 47GP A car with 58-cm-diameter tires accelerates uniformly from rest to 20 m/s in 10 s. How many times does each tire rotate?

Problem 48GP The cable lifting an elevator is wrapped around a 1.0-mdiameter cylinder that is turned by the elevator’s motor. The elevator is moving upward at a speed of 1.6 m/s. It then slows to a stop as the cylinder makes one complete turn at constant angular acceleration. How long does it take for the elevator to stop?

Problem 49GP The 20-cm-diameter disk in Figure P7.59 can rotate on an axle through its center. What is the net torque about the axle?

Problem 50GP A combination lock has a 1.0-cm-diameter knob that is part of the dial you turn to unlock the lock. To turn that knob, you grip it between your thumb and forefinger with a force of 0.60 N as you twist your wrist. Suppose the coefficient of static friction between the knob and your fingers is only 0.12 because some oil accidentally got onto the knob. What is the most torque you can exert on the knob without having it slip between your fingers?

Problem 51GP A 70 kg man’s arm, including the hand, can be modeled as a 75-cm-long uniform cylinder with a mass of 3.5 kg. In raising both his arms, from hanging down to straight up, by how much does he raise his center of gravity?

Problem 52GP A penny has a mass of 2.5 g and is 1.5 mm thick; a nickel has a mass of 5.7 g and is 1.9 mm thick. If you make a stack of coins on a table, starting with five nickels and finishing with four pennies, how far above me tabletop is the center of gravity of the stack?

Problem 53GP The machinist's square shown in Figure 53 consists of a thin, rectangular blade connected to a rectangular handle. a. Determine the x and y coordinates of the center or gravity. Let the lower left corner be x ? 0, y = 0. b. Sketch how the tool would hang if it were allowed to freely pivot about the point x = 0, y = 0. c. When hanging from that point, what angle would the long side of the blade make with the vertical? FIGURE 53

Problem 54GP The four masses shown in Figure 54 are connected by massless, rigid rods. a. Find the coordinates of the center of gravity. b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page. c. Find the moment of inertia about a diagonal axis that passes through masses B and D. FIGURE 54

Problem 55GP Three 0.10 kg balls are connected by light rods to form an equilateral triangle with a side length of 0.30 m. What is the moment of inertia of this triangle about an axis perpendicular to its plane and passing through one of the balls?

Problem 56GP The three masses shown in Figure P7.62 are connected by massless, rigid rods. a. Find the coordinates of the center of gravity. b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page. c. Find the moment of inertia about an axis that passes through masses B and C.

Problem 57GP A reasonable estimate of the moment of inertia of an ice skater spinning with her arms at her sides can be made by modeling most of her body as a uniform cylinder. Suppose the skater has a mass of 64 kg. One-eighth of that mass is in her arms, which are 60 cm long and 20 cm from the vertical axis about which she rotates. The rest of her mass is approximately in the form of a 20-cm-radius cylinder. a. Estimate the skater’s moment of inertia to two significant figures. b. If she were to hold her arms outward, rather than at her sides, would her moment of inertia increase, decrease, or remain unchanged? Explain.

Problem 58GP Starting from rest, a 12-cm-diameter compact disk takes 3.0 s to reach its operating angular velocity of 2000 rpm. Assume that the angular acceleration is constant. The disk’s moment of inertia is . a. How much torque is applied to the disk? b. How many revolutions does it make before reaching full speed?

Problem 59GP The ropes in Figure P7.65 are each wrapped around a cylinder, and the two cylinders are fastened together. The smaller cylinder has a diameter of 10 cm and a mass of 5.0 kg; the larger cylinder has a diameter of 20 cm and a mass of 20 kg. What is the angular acceleration of the cylinders? Assume that the cylinders turn on a frictionless axle.

Problem 60GP Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel’s energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. A motor spins up the flywheel with a constant torque of 50 N . m. How long does it take the flywheel to reach top angular speed of 1200 rpm?

Problem 61GP A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid, massless rod. The rod and balls are rotating clockwise about their center of gravity at 20 rpm. What torque will bring the balls to a halt in 5.0 s?

Problem 62GP A 1.5 kg block is connected by a rope across a 50-cm-diameter, 2.0 kg, frictionless pulley, as shown in Figure P7.68. A constant 10 N tension is applied to the other end of the rope. Starting from rest, how long does it take the block to move 30 cm?

Problem 63GP The two blocks in Figure P7.69 are connected by a massless rope that passes over a pulley. The pulley is 12 cm in diameter and has a mass of 2.0 kg. As the pulley turns, friction at the axle exerts a torque of magnitude 0.50 N . m. If the blocks are released from rest, how long does it take the 4.0 kg block to reach the floor?

Problem 64GP The 2.0 kg, 30-cm-diameter disk in Figure P7.70 is spinning at 300 rpm. How much friction force must the brake apply to the rim to bring the disk to a halt in 3.0 s?

Problem 65GP A tradesman sharpens a knife by pushing it against the rim of a grindstone. The 30-cm-diameter stone is spinning at 200 rpm and has a mass of 28 kg. The coefficient of kinetic friction between the knife and the stone is 0.20. If the stone loses 10% of its speed in 10 s of grinding, what is the force with which the man presses the knife against the stone?

Problem 66GP The bunchberry flower has the fastest-moving parts ever BIO seen in a plant. Initially, the stamens are held by the petals in a bent position, storing elastic energy like a coiled spring. As the petals release, the tips of the stamens act like medieval catapults, flipping through a 60° angle in just 0.30 ms to launch pollen from the anther sacs at their ends. The human eye just sees a burst of pollen; careful photography (see Figure 66a) reveals the details. As shown in Figure 66b, we can model a stamen tip as a 1.0-mm-long, 10/ µg rigid rod with a 10/ µg anther sac at one end and a pivot point at the opposite end. Although oversimplifying, we will assume that the angular acceleration is constant throughout the motion. (a). (b). FIGURE 66 a. What is the tangential acceleration of the anther sac during the motion? b. What is the speed of the anther sac as it releases its pollen? c. How large is the "straightening torque"? Neglect gravitational forces in your calculation. d. Compute the gravitational torque on the stamen tip (including the anther sac) in its initial orientation. Was it reasonable to neglect the gravitational torque in part c?

Problem 67PP The Illusion of Flight The grand jeté is a classic ballet maneuver in which a dancer executes a horizontal leap while moving her arms and legs up and then down. At the center of the leap, the arms and legs are gracefully extended, as we see in Figure P7.75a. The goal of the leap is to create the illusion of flight. As the dancer moves through the air, he or she is in free fall. In Chapter 3, we saw that this leads to projectile motion. But what part of the dancer follows the usual parabolic path? It won’t come as a surprise to learn that it’s the center of gravity. But when you watch a dancer leap through the air, you don’t watch her center of gravity, you watch her head. If the translational motion of her head is horizontal—not parabolic—this creates the illusion that she is flying through the air, held up by unseen forces. Figure P7.75b illustrates how the dancer creates this illusion. While in the air, she changes the position of her center of gravity relative to her body by moving her arms and legs up, then down. Her center of gravity moves in a parabolic path, but her head moves in a straight line. It’s not flight, but it will appear that way, at least for a moment. To perform this maneuver, the dancer relies on the fact that the position of her center of gravity A. Is near the center of the torso. B. Is determined by the positions of her arms and legs. C. Moves in a horizontal path. D. Is outside of her body.

Problem 68PP The Illusion of Flight The grand jeté is a classic ballet maneuver in which a dancer executes a horizontal leap while moving her arms and legs up and then down. At the center of the leap, the arms and legs are gracefully extended, as we see in Figure P7.75a. The goal of the leap is to create the illusion of flight. As the dancer moves through the air, he or she is in free fall. In Chapter 3, we saw that this leads to projectile motion. But what part of the dancer follows the usual parabolic path? It won’t come as a surprise to learn that it’s the center of gravity. But when you watch a dancer leap through the air, you don’t watch her center of gravity, you watch her head. If the translational motion of her head is horizontal—not parabolic—this creates the illusion that she is flying through the air, held up by unseen forces. Figure P7.75b illustrates how the dancer creates this illusion. While in the air, she changes the position of her center of gravity relative to her body by moving her arms and legs up, then down. Her center of gravity moves in a parabolic path, but her head moves in a straight line. It’s not flight, but it will appear that way, at least for a moment. Suppose you wish to make a vertical leap with the goal of getting your head as high as possible above the ground. At the top of your leap, your arms should be A. Held at your sides. B. Raised above your head. C. Outstretched, away from your body.

Problem 69PP The Illusion of Flight The grand jeté is a classic ballet maneuver in which a dancer executes a horizontal leap while moving her arms and legs up and then down. At the center of the leap, the arms and legs are gracefully extended, as we see in Figure P7.75a. The goal of the leap is to create the illusion of flight. As the dancer moves through the air, he or she is in free fall. In Chapter 3, we saw that this leads to projectile motion. But what part of the dancer follows the usual parabolic path? It won’t come as a surprise to learn that it’s the center of gravity. But when you watch a dancer leap through the air, you don’t watch her center of gravity, you watch her head. If the translational motion of her head is horizontal—not parabolic—this creates the illusion that she is flying through the air, held up by unseen forces. Figure P7.75b illustrates how the dancer creates this illusion. While in the air, she changes the position of her center of gravity relative to her body by moving her arms and legs up, then down. Her center of gravity moves in a parabolic path, but her head moves in a straight line. It’s not flight, but it will appear that way, at least for a moment. When the dancer is in the air, is there a gravitational torque on her? Take the dancer’s rotation axis to be through her center of gravity. A. Yes, there is a gravitational torque. B. No, there is not a gravitational torque. C. It depends on the positions of her arms and legs.

Problem 70PP The Illusion of Flight The grand jeté is a classic ballet maneuver in which a dancer executes a horizontal leap while moving her arms and legs up and then down. At the center of the leap, the arms and legs are gracefully extended, as we see in Figure P7.75a. The goal of the leap is to create the illusion of flight. As the dancer moves through the air, he or she is in free fall. In Chapter 3, we saw that this leads to projectile motion. But what part of the dancer follows the usual parabolic path? It won’t come as a surprise to learn that it’s the center of gravity. But when you watch a dancer leap through the air, you don’t watch her center of gravity, you watch her head. If the translational motion of her head is horizontal—not parabolic—this creates the illusion that she is flying through the air, held up by unseen forces. Figure P7.75b illustrates how the dancer creates this illusion. While in the air, she changes the position of her center of gravity relative to her body by moving her arms and legs up, then down. Her center of gravity moves in a parabolic path, but her head moves in a straight line. It’s not flight, but it will appear that way, at least for a moment. In addition to changing her center of gravity, a dancer may change her moment of inertia. Consider her moment of inertia about a vertical axis through the center of her body. When she raises her arms and legs, this A. Increases her moment of inertia. B. Decreases her moment of inertia. C. Does not change her moment of inertia.