Suppose you wish to make a vertical leap with the goal ofgetting your head as high as

The batter in a baseball game hits a home run. As he circles the bases, is his angular velocity positive or negative?

Viewed from somewhere in space above the north pole, would a point on the earth's equator have a posi(ive or negative angular velocity due to the earth's rotation?

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 acce)eration of the pulley in each case?

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 th is would help you loosen the nut.

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?

A screwdriver with a very thick handle requires less FIGURE 07.5 (d) force to operate than one with a very skinny handle. Explain why this is so.

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?

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?

A student gives a steady push to a ball at the end of a massless, rigid rod for I s, causing the ball to rotate clockwise in a horizontal circle as shown in Figure Q7.9. The rod's pivot is friction less. Sketch a graph of the ball's angular velocity as a function of time for the first 3 s of the ball's motion. You won't be able to include numbers on the vertical axis, but your graph should have the correct sign and the correct shape.

You can use a simple technique to find the center of gravity of an irregular shape. Figure Q7. IO shows a cardboard cutout of the outline of the continental United States. The map is suspended from pivot I and allowed to hang freely; then a blue vertical li ne is drawn. The map is then suspended from pivot 2, hangs freely, and the red vertical line is drawn. The center of gravity lies at the intersection of the two lines. Exp)ain how this technique works.

The two ends of the dumbbell shown in Figure Q7. I I are made of the same material. Is the dumbbell' s center of gravity at point I, 2, or 3? Explain.

If you grasp a hammer by its lightweight handle and wave it back and forth, and then grasp it by its much heavier head and wave it back and forth, as in the figure, you' II find that you can wave the hammer much more rapidly in the second case, when you grasp it by the head. Explain why th is is so.

Suppose you have two identical-looking metal spheres of the same size and the same mass. One of them is solid; the other is hollow. Describe a simple test that you could do to determine which is which.

The moment of inertia of a uniform rod about an axis through its center is ML2/l 2. The moment of inertia about an axis at one end is ML2!3. Explain why the moment of inertia is larger about the end than about the center.

The wheel in Figure Q7 .15 is rolling to the right without slipping. Rank in order, from fastest to slowest, the speeds of the points labeled I through 5. Explain your reasoning .

With care, it's possible to walk on top of a barrel as it rolls. It is much easier to do this if the barrel is full than if it is empty. Explain why this is so.

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

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

A machine part is made up of two pieces, with centers of gravity shown in Figure Q7. l 9. Which point could be the center of gravity of the entire part?

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? A. 2.7 X 10- 5 kg m2 B. 5.4 X 10-5 kg m2 C. I.I X I0-4 kg m2 D. 2.2X l0-4 kgm2

Suppose manufacturers increase the size of compact disks so that they are made of the same material and have the same thickness as a current disk but have twice the diameter. By what factor will the moment of inertia increase? A. 2 B. 4 C. 8 D. 16

Two horizontal rods are each held up by vertical strings tied to their ends. Rod I has length Land mass M; rod 2 has length 2L and mass 2M. Each rod then has one of its supporting strings cut, causing the rod to begin pivoting about the end that is still tied up. Which rod has a larger initial angular acceleration? A. Rod 1 B. Rod 2 C. The initial angular acceleration is the same for both.

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.

A paJticle undergoing circular motion in the xy-plane stops on the positive y-axis. Which of the following does not describe its angular position? A. 7T/2 rad B. 7T rad C. 57T/2 rad D. - 37T/2 rad

What is the approximate angular speed of the skater during the leap? A. 2 rad/s B. 6 rad/s C. 9 rad/s D. 20 rad/s

The skater's arms are fully extended during the jump. What is the approximate centripetal acceleration of the skater's hand? A. 10 m/s2 B. 30 m/s2 C. 300 m/s2 D. 450 mls2

What is the approximate speed of the skater's hand? A. I m/s B. 3 mis C. 9 m/s D. 15 mis

Three identical coins lie on three corners of a square I 0.0 cm on a side, as shown in Figure P7.28. Determine the x- and y-coordi nates of the center of gravity of the group of three coins.

Hold your arm outstretched so that it is horizontal. Estimate BIO 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?

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?

The 2.0 kg, uniform, horizontal rod in Figure P7.3 I is seen from the side. What is the gravitational torque about the point shown?

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 underconstmction. A 70.0 kg constmction 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?

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

The 2.0-m-long, 15 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?

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.

Figure P7.36 shows two thin beams joined at tight 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 ax is through the corner. The beams are seen from the side.

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?

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 /1, /2, and /3 about axes through the centers of the rods.

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 ioettia about the rotation axis?

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?

A bicycle rim has a diameter of 0.65 m and a moment of inertia, measured about its center, of 0.19 kg m2 . What is the mass of the rim?

a. What is the moment of inertia of the door in Problem 2 1? b. If you let go of the open door, what is its angular acceleration immediately afterward?

A small grinding wheel has a moment of inertia of 4.0 X I o-5 kg m2. What net torque must be applied to the wheel for its angular acceleration to be 150 rad/s2?

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 1.8 rad/s2 , what is the total moment of inertia about the axis of you and the chair?

An object's moment of inertia is 2.0 kg m2. Its angular velocity is increasing at the rate of 4.0 rad/s per second. What is the net torque on the object?

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?

The 2.5 kg object shown in Figure P7.47 has a moment of inertia about the rotation axis of 0.085 kg m2. The rotation axis is horizontal. When released, what wi ll be the object's initial angular acceleration?

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

Jf you li ft the front wheel of a poorly mai ntai ned 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 0.30 kg m2 , what is the magnitude of the frictional torque?

On page 207 there is a photograph of a girl pushing on a large stone sphere. The sphere has a mass of 8200 kg and a radius of 90 cm. Suppose that she pushes on the sphere tangent to its surface with a steady force of 50 N and that the pressured water provides a frictionless support. How long wi ll it take her to rotate the sphere one ti me, starting from rest?

A toy top with a spool of diameter 5.0 cm has a moment of inertia of 3.0X I 0-5 kg m2 about its rotation axis. To get the top spinning, its string is pulled with a tension of0.30 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.

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?

A bicycle with 0. 80-m-diameter tires is coasting on a level road at 5.6 mis. 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?

A 2.0-m-long slab of concrete is supported by rollers, as shown in Figure P7.54. If the slab is pushed to the right, it will move off the supporting rollers, one by one, on the left side. How far can the slab be moved before its center of gravity is to the right of the contact point with the rightmost roller- at which point the slab begins to tip?

Figure P7.55 shows the angular position-versus-time graph INT for a part icle moving in a c ircle. a. Wri te a description of the particle's motion. b. Draw the angular velocity-versus-time graph.

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.

A car with 58-cm-diameter tires accelerates uniformly fromT rest to 20 m/s in IO s. How many times does each tire rotate?

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 mis. It then slows to a stop as the cylinder makes one complete turn at constant angular acceleration. How long does it take for the e levator to stop?

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?

A combination lock has W 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 20N your thumb and forefinger with a force of 0.60 N as you twist your wri st. Suppose the coeffi cient of static fri ction 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?

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. Jn raising both his arms, from hanging down to straight up, by bow much does he raise his center of gravity?

The three masses shown in Figure P7 .62 are connected by massle ss, rigid rods. a. Find the coordinates of the center of gravity. b. Find the mome nt 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.

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. Tf she were to hold her arms outward, rather than at her sides, would her moment of inertia increase, decrease, or remain unchanged? Explain.

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 2.5 x IQ-5 kg m2 a. How much torque is applied to the disk? b. How many revolutions does it make before reaching full speed?

The ropes in Figure P7.65 are each wrapped around a cylinder, and the two cy Ii nders are fastened together. The smaller cylinder has a diameter of I 0 cm and a mass of 5.0 kg; the larger cylinder has a diameter of 20 cm and a mass of 20 kg. Whal is the angular acceleration of the c ylinders? Assume that the cylinders turn on a frictionless axle.

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 accompli sh 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?

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?

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?

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, fri ction 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?

The 2.0 kg, 30-cm-diameter disk in Figure P7.70 is spinning at 300 rpm. How much fri ction force must the brake apply to the rim to bri ng the disk to a halt in 3.0 s?

A tradesman sharpens a knife by pushing it with a constant force 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 lrnife and the stone is 0.20. If the stone slows steadily to 180 rpm in I 0 s of grinding, what is the force with which the man presses the knife against the stone?

What is the angular acceleration of the anther sac during the motion? A. 3.5 x 103 rad/s2 C. 1.2 X 107 rad/s2 B. 7 .0 x 103 rad/s2 D. 2.3 X I 07 rad/s2

What is the speed of the anther sac as it releases its pollen? A. 3.5 m/s B. 7.0 m/s C. 10 mis D. 14 mi

How large is the "straightening torque"? (You can omit gravitational forces from your calculation; the gravitational torque is much less than this.) A. 2.3X 10- 7 Nm C. 2.3 x 10-5 N m The Illusion of Flight B. 3. 1 x 10-7 N m D. 3.1 x J0-5 N m

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. Ts determined by the positions of her arms and legs. C. Moves in a horizontal path. D. ls outside of her body.

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 (rom your body.

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.

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.