A giant swing at an amusement park consists of a 365-kg

What happens if the woman suddenly slides closer to the hub by 0.400 m?

To make the wedge more effective in keeping the door closed, should it be placed closer to the hinge or to the doorknob?

What happens if the woman now leans backwards?

If 1.00 kg is added to the masses on the left and right, does the center of mass (a) move to the left, (b) move to the right, or (c) remain in the same position.

What would happen if a support is placed exactly at x 79 cm followed by the removal of the supports at the subjects head and feet?

Suppose the biceps were surgically reattached three centimeters farther toward the persons hand. If the same bowling ball were again held in the persons hand, how would the force required of the biceps be affected? Explain.

What happens to the tension in the cable if the man in Figure 8.12a moves farther away from the wall?

If a 50.0 N monkey hangs from the middle rung, would the coeffi cient of static friction be (a) doubled, (b) halved, or (c) unchanged?

If one of the rods is lengthened, which one would cause the larger change in the moment of inertia, the rod connecting bars one and three or the rod connecting balls two and four?

Why do pitchers step forward when delivering the pitch? Why is the timing important?

How would the acceleration and tension change if most of the reels mass were at its rim?

Rank from fastest to slowest: (a) a solid ball rolling down a ramp without slipping, (b) a cylinder rolling down the same ramp without slipping, (c) a block sliding down a frictionless ramp with the same height and slope.

How would increasing the radius of the pulley affect the fi nal answer? Assume the angles of the cables are unchanged and the mass is the same as before.

If the student suddenly releases the weights, will his angular speed increase, decrease, or remain the same?

Is energy conservation violated in this example? Explain why there is a positive net change in mechanical energy. What is the origin of this energy?

A wrench 0.500 m long is applied to a nut with a force of 80.0 N. Because of the cramped space, the force must be exerted upward at an angle of 60.0 with respect to a line from the bolt through the end of the wrench. How much torque is applied to the nut? (a) 34.6 N m (b) 4.56 N m (c) 11.8 N m (d) 14.2 N m (e) 20.0 N m

A horizontal plank 4.00 m long and having mass 20.0 kg rests on two pivots, one at the left end and a second 1.00 m from the right end. Find the magnitude of the force exerted on the plank by the second pivot. (a) 32.0 N (b) 45.2 N (c) 112 N (d) 131 N (e) 98.2 N

What is the magnitude of the angular acceleration of a 25.0-kg disk of radius 0.800 m when a torque of mag ni tude 40.0 N m is applied to it? (a) 2.50 rad/s2 (b) 5.00 rad/s2 (c) 7.50 rad/s2 (d) 10.0 rad/s2 (e) 12.5 rad/s2

Estimate the rotational kinetic energy of Earth by treating it as a solid sphere with uniform density. (a) 3 1029 kg m2/s2 (b) 5 1027 kg m2/s2 (c) 7 1030 kg m2/s2 (d) 4 1028 kg m2/s2 (e) 2 1025 kg m2/s2

Two forces are acting on an object. Which of the following statements is correct? (a) The object is in equilibrium if the forces are equal in magnitude and opposite in direction. (b) The object is in equilibrium if the net torque on the object is zero. (c) The object is in equilibrium if the forces act at the same point on the object. (d) The object is in equilibrium if the net force and the net torque on the object are both zero. (e) The object cannot be in equilibrium because more than one force acts on it.

A disk rotates about a fi xed axis that is perpendicular to the disk and passes through its center. At any instant, does every point on the disk have the same (a) centripetal acceleration, (b) angular velocity, (c) tangential acceleration, (d) linear velocity, or (e) total acceleration?

A constant net nonzero torque is exerted on an object. Which of the following quantities cannot be constant for this object? More than one answer may be correct. (a) angular acceleration (b) angular velocity (c) moment of inertia (d) center of mass (e) angular momentum

A block slides down a frictionless ramp, while a hollow sphere and a solid ball roll without slipping down a second ramp with the same height and slope. Rank the arrival times at the bottom from shortest to longest. (a) sphere, ball, block (b) ball, block, sphere (c) ball, sphere, block (d) block, sphere, ball (e) block, ball, sphere

A solid disk and a hoop are simultaneously released from rest at the top of an incline and roll down without slipping. Which object reaches the bottom fi rst? (a) The one that has the largest mass arrives fi rst. (b) The one that has the largest radius arrives fi rst. (c) The hoop arrives fi rst. (d) The disk arrives fi rst. (e) The hoop and the disk arrive at the same time.

A solid cylinder of mass M and radius R rolls down an incline without slipping. Its moment of inertia about an axis through its center of mass is MR2 /2. At any instant while in motion, its rotational kinetic energy about its center of mass is what fraction of its total kinetic energy? (a) 1 2 (b) 1 4 (c) 1 3 (d) 2 5 (e) None of these

The cars in a soapbox derby have no engines; they simply coast downhill. Which of the following design criteria is best from a competitive point of view? The cars wheels should (a) have large moments of inertia, (b) be massive, (c) be hoop-like wheels rather than solid disks, (d) be large wheels rather than small wheels, or (e) have small moments of inertia.

Consider two uniform, solid spheres, a large, massive sphere and a smaller, lighter sphere. They are released from rest simultaneously from the top of a hill and roll down without slipping. Which one reaches the bottom of the hill fi rst? (a) The large sphere reaches the bottom fi rst. (b) The small sphere reaches the bottom fi rst. (c) The sphere with the greatest density reaches the bottom fi rst. (d) The spheres reach the bottom at the same time. (e) The answer depends on the values of the spheres masses and radii.

A mouse is initially at rest on a horizontal turntable mounted on a frictionless, vertical axle. As the mouse begins to walk clockwise around the perimeter, which of the following statements must be true of the turntable? (a) It also turns clockwise. (b) It turns counterclockwise with the same angular velocity as the mouse. (c) It remains stationary. (d) It turns counterclockwise because angular momentum is conserved. (e) It turns counterclockwise because mechanical energy is conserved. CON

Why cant you put your heels fi rmly against a wall and then bend over without falling?

Why does a tall athlete have an advantage over a smaller one when the two are competing in the high jump?

Both torque and work are products of force and distance. How are they different? Do they have the same units?

Is it possible to calculate the torque acting on a rigid object without specifying an origin? Is the torque independent of the location of the origin?

Can an object be in equilibrium when only one force acts on it? If you believe the answer is yes, give an example to support your conclusion.

In the movie Jurassic Park, there is a scene in which some members of the visiting group are trapped in the kitchen with dinosaurs outside. The paleontologist is pressing against the center of the door, trying to keep out the dinosaurs on the other side. The botanist throws herself against the door at the edge near the hinge. A pivotal point in the fi lm is that she cannot reach a gun on the fl oor because she is trying to hold the door closed. If the paleontologist is pressing at the center of the door, and the botanist is pressing at the edge about 8 cm from the hinge, estimate how far the paleontologist would have to relocate in order to have a greater effect on keeping the door closed than both of them pushing together have in their original positions. (Question 6 is courtesy of Edward F. Redish. For more questions of this type, see www.physics.umd .edu/perg/.)

In some motorcycle races, the riders drive over small hills and the motorcycle becomes airborne for a short time. If the motorcycle racer keeps the throttle open while leav9ing the hill and going into the air, the motorcycles nose tends to rise upwards. Why does this happen?

If you toss a textbook into the air, rotating it each time about one of the three axes perpendicular to it, you will fi nd that it will not rotate smoothly about one of those axes. (Try placing a strong rubber band around the book before the toss so that it will stay closed.) The books rotation is stable about those axes having the largest and smallest moments of inertia, but unstable about the axis of intermediate moment. Try this on your own to fi nd the axis that has this intermediate moment of inertia.

Stars originate as large bodies of slowly rotating gas. Because of gravity, these clumps of gas slowly decrease in size. What happens to the angular speed of a star as it shrinks? Explain.

If a high jumper positions his body correctly when going over the bar, the center of gravity of the athlete may actually pass under the bar. (See Fig. CQ8.10.) Explain how this is possible

In a tape recorder, the tape is pulled past the readwrite heads at a constant speed by the drive mechanism. Consider the reel from which the tape is pulled: As the tape is pulled off, the radius of the roll of remaining tape decreases. How does the torque on the reel change with time? How does the angular speed of the reel change with time? If the tape mechanism is suddenly turned on so that the tape is quickly pulled with a large force, is the tape more likely to break when pulled from a nearly full reel or from a nearly empty reel?

(a) Give an example in which the net force acting on an object is zero, yet the net torque is nonzero. (b) Give an example in which the net torque acting on an object is zero, yet the net force is nonzero.

A ladder rests inclined against a wall. Would you feel safer climbing up the ladder if you were told that the fl oor was frictionless, but the wall was rough, or that the wall was frictionless, but the fl oor was rough? Justify your answer.

A cat usually lands on its feet regardless of the position from which it is dropped. A slow-motion fi lm of a cat falling shows that the upper half of its body twists in one direction while the lower half twists in the opposite direction. (See Fig. CQ8.14.) Why does this type of rotation occur?

A grinding wheel of radius 0.350 m rotating on a frictionless axle is brought to rest by applying a constant friction force tangential to its rim. The constant torque produced by this force is 76.0 N  m. Find the magnitude of the friction force.

According to the manual of a certain car, a maximum torque of magnitude 65.0 N m should be applied when tightening the lug nuts on the vehicle. If you use a wrench of length 0.330 m and you apply the force at the end of the wrench at an angle of 75.0 with respect to a line going from the lug nut through the end of the handle, what is the magnitude of the maximum force you can exert on the handle without exceeding the recommendation?

Calculate the net torque (magnitude and direction) on the beam in Figure P8.3 about (a) an axis through O perpendicular to the page and (b) an axis through C perpendicular to the page.

A steel band exerts a horizontal force of 80.0 N on a tooth at point B in Figure P8.4. What is the torque on the root of the tooth about point A?

A simple pendulum consists of a small object of mass 3.0 kg hanging at the end of a 2.0-m-long light string that is connected to a pivot point. (a) Calculate the magnitude of the torque (due to the force of gravity) about this pivot point when the string makes a 5.0 angle with the vertical. (b) Does the torque increase or decrease as the angle increases? Explain.

Write the necessary equations of equilibrium of the object shown in Figure P8.6. Take the origin of the torque equation about an axis perpendicular to the page through the point O.

The arm in Figure P8.7 weighs 41.5 N. The force of gravity acting on the arm acts through point A. Determine the magnitudes of the tension force F S t in the deltoid muscle and the force F S s exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown.

A uniform beam of length 7.60 m and weight 4.50 102 N is carried by two workers, Sam and Joe, as shown in Figure P8.8. (a) Determine the forces that each person exerts on the beam. (b) Qualitatively, how would the answers change if Sam moved closer to the midpoint? (c) What would happen if Sam moved beyond the midpoint?

A cook holds a 2.00-kg carton of milk at arms length (Fig. P8.9). What force F S B must be exerted by the biceps muscle? (Ignore the weight of the forearm.)

A meterstick is found to balance at the 49.7-cm mark when placed on a fulcrum. When a 50.0-gram mass is attached at the 10.0-cm mark, the fulcrum must be moved to the 39.2-cm mark for balance. What is the mass of the meterstick?

Find the x- and y-coordinates of the center of gravity of a 4.00-ft by 8.00-ft uniform sheet of plywood with the upper right quadrant removed as shown in Figure P8.11.

beam resting on two pivots has a length of L 6.00 m and mass M 90.0 kg. The pivot under the left end exerts a normal force n1 on the beam, and the second pivot placed a distance 4.00 m from the left end exerts a normal force n2. A woman of mass m 55.0 kg steps onto the left end of the beam and begins walking to the right as in Figure P8.12. The goal is to fi nd the womans position when the beam begins to tip. (a) Sketch a free-body diagram, labeling the gravitational and normal forces acting on the beam and placing the woman x meters to the right of the fi rst pivot, which is the origin. (b) Where is the woman when the normal force n1 is the greatest? (c) What is n1 when the beam is about to tip? (d) Use the force equation of equilibrium to fi nd the value of n2 when the beam is about to tip. (e) Using the result of part (c) and the torque equilibrium equation, with torques computed around the second pivot point, fi nd the womans position when the beam is about to tip. (f) Check the answer to part (e) by computing torques around the fi rst pivot point. Except for possible slight differences due to rounding, is the answer the same?

Consider the following mass distribution, where x- and y-coordinates are given in meters: 5.0 kg at (0.0, 0.0) m, 3.0 kg at (0.0, 4.0) m, and 4.0 kg at (3.0, 0.0) m. Where should a fourth object of 8.0 kg be placed so that the center of gravity of the four-object arrangement will be at (0.0, 0.0) m? 14. ecp A beam of length L and mass M rests on two pivots. The fi rst pivot is at the left end, taken as the origin, and the second pivot is at a distance from the left end. A woman of mass m starts at the left end and walks toward the right end as in Figure P8.12. When the beam is on the verge of tipping, fi nd symbolic expressions for (a) the normal force exerted by the second pivot in terms of M, m, and g and (b) the womans position in terms of M, m, L, and . (c) Find the minimum value of that will allow the woman to reach the end of the beam without it tipping.

A grinding wheel of radius 0.350 m rotating on a frictionless axle is brought to rest by applying a constant friction force tangential to its rim. The constant torque produced by this force is 76.0 N m. Find the magnitude of the friction force.

According to the manual of a certain car, a maximum torque of magnitude 65.0 N m should be applied when tightening the lug nuts on the vehicle. If you use a wrench of length 0.330 m and you apply the force at the end of the wrench at an angle of 75.0 with respect to a line going from the lug nut through the end of the handle, what is the magnitude of the maximum force you can exert on the handle without exceeding the recommendation?

Calculate the net torque (magnitude and direction) on the beam in Figure P8.3 about (a) an axis through O perpendicular to the page and (b) an axis through C perpendicular to the page.

A steel band exerts a horizontal force of 80.0 N on a tooth at point B in Figure P8.4. What is the torque on the root of the tooth about point A? FIGURE P8.6 Fx Fy Rx O Ry Fg u FIGURE P8.4 B 48.0 1.20 cm A Gum

A simple pendulum consists of a small object of mass 3.0 kg hanging at the end of a 2.0-m-long light string that is connected to a pivot point. (a) Calculate the magnitude of the torque (due to the force of gravity) about this pivot point when the string makes a 5.0 angle with the vertical. (b) Does the torque increase or decrease as the angle increases? Explain.

Write the necessary equations of equilibrium of the object shown in Figure P8.6. Take the origin of the torque equation about an axis perpendicular to the page through the point O.

The arm in Figure P8.7 weighs 41.5 N. The force of gravity acting on the arm acts through point A. Determine the magnitudes of the tension force F S t in the deltoid muscle and the force F S s exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown.

A uniform beam of length 7.60 m and weight 4.50 102 N is carried by two workers, Sam and Joe, as shown in Figure P8.8. (a) Determine the forces that each person exerts on the beam. (b) Qualitatively, how would the answers change if Sam moved closer to the midpoint? (c) What would happen if Sam moved beyond the midpoint? 9. A cook holds a 2.00-kg carton of milk at arms length (Fig. P8.9). What force F S B must be exerted by the biceps muscle? (Ignore the weight of the forearm.)

A cook holds a 2.00-kg carton of milk at arms length (Fig. P8.9). What force F S B must be exerted by the biceps muscle? (Ignore the weight of the forearm.) at the 10.0-cm mark, the fulcrum must be moved to the 39.2-cm mark for balance. What is the mass of the meterstick?

Find the x- and y-coordinates of the center of gravity of a 4.00-ft by 8.00-ft uniform sheet of plywood with the upper right quadrant removed as shown in Figure P8.11.

A beam resting on two pivots has a length of L 6.00 m and mass M 90.0 kg. The pivot under the left end exerts a normal force n1 on the beam, and the second pivot placed a distance 4.00 m from the left end exerts a normal force n2. A woman of mass m 55.0 kg steps onto the left end of the beam and begins walking to the right as in Figure P8.12. The goal is to fi nd the womans position when the beam begins to tip. (a) Sketch a free-body diagram, labeling the gravitational and normal forces acting on the beam and placing the woman x meters to the right of the fi rst pivot, which is the origin. (b) Where is the woman when the normal force n1 is the greatest? (c) What is n1 when the beam is about to tip? (d) Use the force equation of equilibrium to fi nd the value of n2 when the beam is about to tip. (e) Using the result of part (c) and the torque equilibrium equation, with torques computed around the second pivot point, fi nd the womans position when the beam is about to tip. (f) Check the answer to part (e) by computing torques around the fi rst pivot point. Except for possible slight differences due to rounding, is the answer the same? FIGU

Consider the following mass distribution, where x- and y-coordinates are given in meters: 5.0 kg at (0.0, 0.0) m, 3.0 kg at (0.0, 4.0) m, and 4.0 kg at (3.0, 0.0) m. Where should a fourth object of 8.0 kg be placed so that the center of gravity of the four-object arrangement will be at (0.0, 0.0) m?

A beam of length L and mass M rests on two pivots. The fi rst pivot is at the left end, taken as the origin, and the second pivot is at a distance from the left end. A woman of mass m starts at the left end and walks toward the right end as in Figure P8.12. When the beam is on the verge of tipping, fi nd symbolic expressions for (a) the normal force exerted by the second pivot in terms of M, m, and g and (b) the womans position in terms of M, m, L, and . (c) Find the minimum value of that will allow the woman to reach the end of the beam without it tipping.

Many of the elements in horizontal-bar exercises can be modeled by representing the gymnast by four segments consisting of arms, torso (including the head), thighs, and lower legs, as shown in Figure P8.15a. Inertial parameters for a particular gymnast are as follows: Segment Mass (kg) Length (m) rcg (m) I (kg m2) Arms 6.87 0.548 0.239 0.205 Torso 33.57 0.601 0.337 1.610 Thighs 14.07 0.374 0.151 0.173 Legs 7.54 0.227 0.164 Note that in Figure P8.15a rcg is the distance to the center of gravity measured from the joint closest to the bar and the masses for the arms, thighs, and legs include both appendages. I is the moment of inertia of each segment about its center of gravity. Determine the distance from the bar to the center of gravity of the gymnast for the two positions shown in Figures P8.15b and P8.15c.

Using the data given in Problem 15 and the coordinate system shown in Figure P8.16b, calculate the position of the center of gravity of the gymnast shown in Figure P8.16a. Pay close attention to the defi nition of rcg in the table.

A person bending forward to lift a load with his back (Fig. P8.17a) rather than with his knees can be injured by large forces exerted on the muscles and vertebrae. The spine pivots mainly at the fi fth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, and to understand why back problems are common among humans, consider the model shown in Figure P8.17b of a person bending forward to lift a 200-N object. The spine and upper body are represented as a uniform horizontal rod of weight 350 N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two-thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is 12.0. Find the tension in the back muscle and the compressional force in the spine.

When a person stands on tiptoe (a strenuous position), the position of the foot is as shown in Figure P8.18a. The total gravitational force on the body, F S g , is supported by the force n S exerted by the fl oor on the toes of one foot. A mechanical model of the situation is shown in Figure P8.18b, where T S is the force exerted by the Achilles tendon on the foot and R S is the force exerted by the tibia on the foot. Find the values of T, R, and u when Fg 700 N.

A 500-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 100-N rod as indicated in Figure P8.19. The left end of the rod is supported by a hinge, and the right end is supported by a thin cable making a 30.0 angle with the vertical. (a) Find the tension T in the cable. (b) Find the horizontal and vertical components of force exerted on the left end of the rod by the hinge.

A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs 200 N and is 3.00 m long. What is the tension in each rope when the 700-N worker stands 1.00 m from one end?

A uniform plank of length 2.00 m and mass 30.0 kg is supported by three ropes, as indicated by the blue vectors in Figure P8.21. Find the tension in each rope when a 700-N person is 0.500 m from the left end.

A hungry 700-N bear walks out on a beam in an attempt to retrieve some goodies hanging at the end (Fig. P8.22). The beam is uniform, weighs 200 N, and is 6.00 m long; the goodies weigh 80.0 N. (a) Draw a free-body diagram of the beam. (b) When the bear is at x 1.00 m, fi nd the tension in the wire and the components of the reaction force at the hinge. (c) If the wire can withstand a maximum tension of 900 N, what is the maximum distance the bear can walk before the wire breaks?

An 8.00-m, 200-N uniform ladder rests against a smooth wall. The coeffi cient of static friction between the ladder and the ground is 0.600, and the ladder makes a 50.0 angle with the ground. How far up the ladder can an 800-N person climb before the ladder begins to slip?

A strut of length L 3.00 m and mass m 16.0 kg is held by a cable at an angle of u 30.0 with respect to the horizontal as shown in Figure P8.24. (a) Sketch a free-body diagram, indicating all the forces and their placement on the strut. (b) Why is the hinge a good place to use for calculating torques? (c) Write the condition for rotational equilibrium symbolically, calculating the torques around the hinge. (d) Use the torque equation to calculate the tension in the cable. (e) Write the x- and y-components of Newtons second law for equilibrium. (f) Use the force equation to fi nd the x- and y-components of the force on the hinge. (g) Assuming the strut position is to remain the same, would it be advantageous to attach the cable higher up on the wall? Explain the benefi t in terms of the force on the hinge and cable tension.

A student gets his car stuck in a snowdrift. Not at a loss, having studied physics, he attaches one end of a stout rope to the car and the other end to the trunk of a nearby tree, allowing for a small amount of slack. The student then exerts a force F S on the center of the rope in the direction perpendicular to the car-tree line as shown in Figure P8.25. If the rope is inextensible and the magnitude of the applied force is 475 N, what is the force on the car? (Assume equilibrium conditions.)

A uniform beam of length L and mass m shown in Figure P8.26 is inclined at an angle u to the horizontal. Its upper end is connected to a wall by a rope, and its lower end rests on a rough horizontal surface. The coeffi cient of static friction between the beam and surface is ms. Assume the angle is such that the static friction force is at its maximum value. (a) Draw a free-body diagram for the beam. (b) Using the condition of rotational equilibrium, fi nd an expression for the tension T in the rope in terms of m, g, and u. (c) Using Newtons second law for equilibrium, fi nd a second expression for T in terms of ms, m, and g. (d) Using the foregoing results, obtain a relationship involving only ms and the angle u. (e) What happens if the angle gets smaller? Is this equation valid for all values of u? Explain.

The chewing muscle, the masseter, is one of the strongest in the human body. It is attached to the mandible (lower jawbone) as shown in Figure P8.27a. The jawbone is pivoted about a socket just in front of the auditory canal. The forces acting on the jawbone are equivalent to those acting on the curved bar in Figure P8.27b. F S c is the force exerted by the food being chewed against the jawbone, T S is the force of tension in the masseter, and R S is the force exerted by the socket on the mandible. Find T S and R S for a person who bites down on a piece of steak with a force of 50.0 N.

A 1 200-N uniform boom is supported by a cable perpendicular to the boom as in Figure P8.28. The boom is hinged at the bottom, and a 2 000-N weight hangs from its top. Find the tension in the supporting cable and the components of the reaction force exerted on the boom by the hinge.

The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Fig. P8.29a). The forces on the lower leg when the leg is extended are modeled as in Figure P8.29b, where T S is the force of tension in the tendon, w S is the force of gravity acting on the lower leg, and F S is the force of gravity acting on the foot. Find T S when the tendon is at an angle of 25.0 with the tibia, assuming that w 30.0 N, F 12.5 N, and the leg is extended at an angle u of 40.0 with the vertical. Assume that the center of gravity of the lower leg is at its center and that the tendon attaches to the lower leg at a point one-fi fth of the way down the leg.

One end of a uniform 4.0-mlong rod of weight w is supported by a cable. The other end rests against a wall, where it is held by friction. (See Fig. P8.30.) The coeffi cient of static friction between the wall and the rod is ms 0.50. Determine the minimum distance x from point A at which an additional weight w (the same as the weight of the rod) can be hung without causing the rod to slip at point A.

Four objects are held in position at the corners of a rectangle by light rods as shown in Figure P8.31. Find the moment of inertia of the system about (a) the x-axis, (b) the y-axis, and (c) an axis through O and perpendicular to the page.

If the system shown in Figure P8.31 is set in rotation about each of the axes mentioned in Problem 30, fi nd the torque that will produce an angular acceleration of 1.50 rad/s2 in each case.

A large grinding wheel in the shape of a solid cylinder of radius 0.330 m is free to rotate on a frictionless, vertical axle. A constant tangential force of 250 N applied to its edge causes the wheel to have an angular acceleration of 0.940 rad/s2. (a) What is the moment of inertia of the wheel? (b) What is the mass of the wheel? (c) If the wheel starts from rest, what is its angular velocity after 5.00 s have elapsed, assuming the force is acting during that time?

An oversized yo-yo is made from two identical solid disks each of mass M 2.00 kg and radius R 10.0 cm. The two disks are joined by a solid cylinder of radius r 4.00 cm and mass m 1.00 kg as in Figure P8.34. Take the center of the cylinder as the axis of the system, with positive torques directed to the left along this axis. All torques and angular variables are to be calculated around this axis. Light string is wrapped around the cylinder, and the system is then allowed to drop from rest. (a) What is the moment of inertia of the system? Give a symbolic answer. (b) What torque does gravity exert on the system with respect to the given axis? (c) Take downward as the negative coordinate direction. As depicted in Figure P8.34, is the torque exerted by the tension positive or negative? Is the angular acceleration positive or negative? What about the translational acceleration? (d) Write an equation for the angular acceleration a in terms of the translational acceleration a and radius r. (Watch the sign!) (e) Write Newtons second law for the system in terms of m, M, a, T, and g. (f) Write Newtons second law for rotation in terms of I, a, T, and r. (g) Eliminate a from the rotational second law with the expression found in part (d) and fi nd a symbolic expression for the acceleration a in terms of m, M, g, r and R. (h) What is the n numeric value for the systems acceleration? (i) What is the tension in the string? (j) How long does it take the system to drop 1.00 m from rest?

A rope of negligible mass is wrapped around a 225-kg solid cylinder of radius 0.400 m. The cylinder is suspended several meters off the ground with its axis oriented horizontally, and turns on that axis without friction. (a) If a 75.0-kg man takes hold of the free end of the rope and falls under the force of gravity, what is his acceleration? (b) What is the angular acceleration of the cylinder? (c) If the mass of the rope were not neglected, what would happen to the angular acceleration of the cylinder as the man falls?

A potters wheel having a radius of 0.50 m and a moment of inertia of 12 kg m2 is rotating freely at 50 rev/min. The potter can stop the wheel in 6.0 s by pressing a wet rag against the rim and exerting a radially inward force of 70 N. Find the effective coeffi cient of kinetic friction between the wheel and the wet rag.

A model airplane with mass 0.750 kg is tethered by a wire so that it fl ies in a circle 30.0 m in radius. The airplane engine provides a net thrust of 0.800 N perpendicular to the tethering wire. (a) Find the torque the net thrust produces about the center of the circle. (b) Find the angular acceleration of the airplane when it is in level fl ight. (c) Find the linear acceleration of the airplane tangent to its fl ight path.

A bicycle wheel has a diameter of 64.0 cm and a mass of 1.80 kg. Assume that the wheel is a hoop with all the mass concentrated on the outside radius. The bicycle is placed on a stationary stand, and a resistive force of 120 N is applied tangent to the rim of the tire. (a) What force must be applied by a chain passing over a 9.00-cmdiameter sprocket in order to give the wheel an acceleration of 4.50 rad/s2? (b) What force is required if you shift to a 5.60-cm-diameter sprocket?

A 150-kg merry-go-round in the shape of a uniform, solid, horizontal disk of radius 1.50 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force must be exerted on the rope to bring the merry-go-round from rest to an angular speed of 0.500 rev/s in 2.00 s?

An Atwoods machine consists of blocks of masses m1 10.0 kg and m2 20.0 kg attached by a cord running over a pulley as in Figure P8.40. The pulley is a solid cylinder with mass M 8.00 kg and radius r = 0.200 m. The block of mass m2 is allowed to drop, and the cord turns the pulley without slipping. (a) Why must the tension T2 be greater than the tension T1? (b) What is the acceleration of the system, assuming the pulley axis is frictionless? (c) Find the tensions T1 and T2.

An airliner lands with a speed of 50.0 m/s. Each wheel of the plane has a radius of 1.25 m and a moment of inertia of 110 kg m2. At touchdown, the wheels begin to spin under the action of friction. Each wheel supports a weight of 1.40 104 N, and the wheels attain their angular speed in 0.480 s while rolling without slipping. What is the coeffi cient of kinetic friction between the wheels and the runway? Assume that the speed of the plane is constant.

A car is designed to get its energy from a rotating fl ywheel with a radius of 2.00 m and a mass of 500 kg. Before a trip, the fl ywheel is attached to an electric motor, which brings the fl ywheels rotational speed up to 5 000 rev/min. (a) Find the kinetic energy stored in the fl ywheel. (b) If the fl ywheel is to supply energy to the car as a 10.0-hp motor would, fi nd the length of time the car could run before the fl ywheel would have to be brought back up to speed.

A horizontal 800-N merry-go-round of radius 1.50 m is started from rest by a constant horizontal force of 50.0 N applied tangentially to the merry-go-round. Find the kinetic energy of the merry-go-round after 3.00 s. (Assume it is a solid cylinder.)

Four objectsa hoop, a solid cylinder, a solid sphere, and a thin, spherical shelleach has a mass of 4.80 kg and a radius of 0.230 m. (a) Find the moment of inertia for each object as it rotates about the axes shown in Table 8.1. (b) Suppose each object is rolled down a ramp. Rank the translational speed of each object from highest to lowest. (c) Rank the objects rotational kinetic energies from highest to lowest as the objects roll down the ramp.

A light rod 1.00 m in length rotates about an axis perpendicular to its length and passing through its center as in Figure P8.45. Two particles of masses 4.00 kg and 3.00 kg are connected to the ends of the rod. (a) Neglect ing the mass of the rod, what is the systems kinetic energy when its angular speed is 2.50 rad/s? (b) Repeat the problem, assuming the mass of the rod is taken to be 2.00 kg.

A 240-N sphere 0.20 m in radius rolls without slipping 6.0 m down a ramp that is inclined at 37 with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest?

A solid, uniform disk of radius 0.250 m and mass 55.0 kg rolls down a ramp of length 4.50 m that makes an angle of 15.0 with the horizontal. The disk starts from rest from the top of the ramp. Find (a) the speed of the disks center of mass when it reaches the bottom of the ramp and (b) the angular speed of the disk at the bottom of the ramp.

A solid uniform sphere of mass m and radius R rolls without slipping down an incline of height h. (a) What forms of mechanical energy are associated with the sphere at any point along the incline when its angular speed is v? Answer in words and symbolically in terms of the quantities m, g, y, I, v, and v. (b) What force acting on the sphere causes it to roll rather than slip down the incline? (c) Determine the ratio of the spheres rotational kinetic energy to its total kinetic energy at any instant.

The top in Figure P8.49 has a moment of inertia of 4.00 10 4 kg m2 and is initially at rest. It is free to rotate about a stationary axis AA . A string wrapped around a peg along the axis of the top is pulled in such a manner as to maintain a con stant tension of 5.57 N in the string. If the string does not slip while wound around the peg, what is the angular speed of the top after 80.0 cm of string has been pulled off the peg? Hint: Consider the work that is done.

. A constant torque of 25.0 N m is applied to a grindstone whose moment of inertia is 0.130 kg m2. Using energy principles and neglecting friction, fi nd the angular speed after the grindstone has made 15.0 revolutions. Hint: The angular equivalent of Wnet F x 1 2mvf 2 2 1 2mvi 2 is Wnet t u 1 2Ivf 2 2 1 2Ivi 2 . You should convince yourself that this relationship is correct.

A 10.0-kg cylinder rolls without slipping on a rough surface. At an instant when its center of gravity has a speed of 10.0 m/s, determine (a) the translational kinetic energy of its center of gravity, (b) the rotational kinetic energy about its center of gravity, and (c) its total kinetic energy.

Use conservation of energy to determine the angular speed of the spool shown in Figure P8.52 after the 3.00-kg bucket has fallen 4.00 m, starting from rest. The light string attached to the bucket is wrapped around the spool and does not slip as it unwinds.

A giant swing at an amusement park consists of a 365-kg uniform arm 10.0 m long, with two seats of negligible mass connected at the lower end of the arm (Fig. P8.53). (a) How far from the upper end is the center of mass of the arm? (b) The gravitational potential energy of the arm is the same as if all its mass were concentrated at the center of mass. If the arm is raised through a 45.0 angle, fi nd the gravitational potential energy, where the zero level is taken to be 10.0 m below the axis. (c) The arm drops from rest from the position described in part (b). Find the gravitational potential energy of the system when it reaches the vertical orientation. (d) Find the speed of the seats at the bottom of the swing.

Each of the following objects has a radius of 0.180 m and a mass of 2.40 kg, and each rotates about an axis through its center (as in Table 8.1) with an angular speed of 35.0 rad/s. Find the magnitude of the angular momentum of each object. (a) a hoop (b) a solid cylinder (c) a solid sphere (d) a hollow spherical shell

(a) Calculate the angular momentum of Earth that arises from its spinning motion on its axis, treating Earth as a uniform solid sphere. (b) Calculate the angular momentum of Earth that arises from its orbital motion about the Sun, treating Earth as a point particle.

A 0.005-kg bullet traveling horizontally with a speed of 1.00 103 m/s enters an 18.0-kg door, imbedding itself 10.0 cm from the side opposite the hinges as in Figure P8.56. The 1.00-m-wide door is free to swing on its hinges. (a) Before it hits the door, does the bullet have angular momentum relative the doors axis of rotation? Explain. (b) Is mechanical energy conserved in this collision? Answer without doing a calculation. (c) At what angular speed does the door swing open immediately after the collision? (The door has the same moment of inertia as a rod with axis at one end.) (d) Calculate the energy of the doorbullet system and determine whether it is less than or equal to the kinetic energy of the bullet before the collision.

A light rigid rod 1.00 m in length rotates about an axis perpendicular to its length and through its center, as shown in Figure P8.45. Two particles of masses 4.00 kg and 3.00 kg are connected to the ends of the rod. What is the angular momentum of the system if the speed of each particle is 5.00 m/s? (Neglect the rods mass.)

Halleys comet moves about the Sun in an elliptical orbit, with its closest approach to the Sun being 0.59 A.U. and its greatest distance being 35 A.U. (1 A.U. is the Earth Sun distance). If the comets speed at closest approach is 54 km/s, what is its speed when it is farthest from the Sun? You may neglect any change in the comets mass and assume that its angular momentum about the Sun is conserved.

The system of small objects shown in Figure P8.59 is rotating at an angular speed of 2.0 rev/s. The objects are connected by light, fl exible spokes that can be lengthened or shortened. What is the new angular speed if the spokes are shortened to 0.50 m? (An effect similar to that illustrated in this problem occurred in the early stages of the formation of our galaxy. As the massive cloud of dust and gas that was the source of the stars and planets contracted, an initially small angular speed increased with time.)

A playground merry-go-round of radius 2.00 m has a moment of inertia I 275 kg m2 and is rotating about a frictionless vertical axle. As a child of mass 25.0 kg stands at a distance of 1.00 m from the axle, the system (merrygo-round and child) rotates at the rate of 14.0 rev/min. The child then proceeds to walk toward the edge of the merry-go-round. What is the angular speed of the system when the child reaches the edge?

A solid, horizontal cylinder of mass 10.0 kg and radius 1.00 m rotates with an angular speed of 7.00 rad/s about a fi xed vertical axis through its center. A 0.250-kg piece of putty is dropped vertically onto the cylinder at a point 0.900 m from the center of rotation and sticks to the cylinder. Determine the fi nal angular speed of the system.

A student sits on a rotating stool holding two 3.0-kg objects. When his arms are extended horizontally, the objects are 1.0 m from the axis of rotation and he rotates with an angular speed of 0.75 rad/s. The moment of inertia of the student plus stool is 3.0 kg m2 and is assumed to be constant. The student then pulls in the objects horizontally to 0.30 m from the rotation axis. (a) Find the new angular speed of the student. (b) Find the kinetic energy of the student before and after the objects are pulled in.

The puck in Figure P8.63 has a mass of 0.120 kg. Its original distance from the center of rotation is 40.0 cm, and it moves with a speed of 80.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. Determine the work done on the puck. Hint: Consider the change in kinetic energy of the puck.

A space station shaped like a giant wheel has a radius of 100 m and a moment of inertia of 5.00 108 kg m2. A crew of 150 lives on the rim, and the station is rotating so that the crew experiences an apparent acceleration of 1g (Fig. P8.64). When 100 people move to the center of the station for a union meeting, the angular speed changes. What apparent acceleration is experienced by the managers remaining at the rim? Assume the average mass of a crew member is 65.0 kg.

A cylinder with moment of inertia I1 rotates with angular velocity v0 about a frictionless vertical axle. A second cylinder, with moment of inertia I2, initially not rotating, drops onto the fi rst cylinder (Fig. P8.65). Because the surfaces are rough, the two cylinders eventually reach the same angular speed v. (a) Calculate v. (b) Show that kinetic energy is lost in this situation, and calculate the ratio of the fi nal to the initial kinetic energy.

A merry-go-round rotates at the rate of 0.20 rev/s with an 80-kg man standing at a point 2.0 m from the axis of rotation. (a) What is the new angular speed when the man walks to a point 1.0 m from the center? Assume that the merry-go-round is a solid 25-kg cylinder of radius 2.0m. (b) Calculate the change in kinetic energy due to the mans movement. How do you account for this change in kinetic energy?

A 60.0-kg woman stands at the rim of a horizontal turntable having a moment of inertia of 500 kg m2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to Earth. (a) In what direction and with what angular speed does the turntable rotate? (b) How much work does the woman do to set herself and the turntable into motion?

Figure P8.68 shows a clawhammer as it is being used to pull a nail out of a horizontal board. If a force of magnitude 150 N is exerted horizontally as shown, fi nd (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface at the point of contact with the hammer head. Assume that the force the hammer exerts on the nail is parallel to the nail and perpendicular to the position vector from the point of contact.

A 40.0-kg child stands at one end of a 70.0-kg boat that is 4.00 m long (Fig. P8.69). The boat is initially 3.00 m from the pier. The child notices a turtle on a rock beyond the far end of the boat and proceeds to walk to that end to catch the turtle. (a) Neglecting friction between the boat and water, describe the motion of the system (child plus boat). (b) Where will the child be relative to the pier when he reaches the far end of the boat? (c) Will he catch the turtle? (Assume that he can reach out 1.00 m from the end of the boat.)

A 12.0-kg object is attached to a cord that is wrapped around a wheel of radius r 10.0 cm (Fig. P8.70). The acceleration of the object down the frictionless incline is measured to be 2.00 m/s2. Assuming the axle of the wheel to be frictionless, determine (a) the tension in the rope, (b) the moment of inertia of the wheel, and (c) the angular speed of the wheel 2.00 s after it begins rotating, starting from rest.

A uniform ladder of length L and weight w is leaning against a vertical wall. The coeffi cient of static friction between the ladder and the fl oor is the same as that between the ladder and the wall. If this coeffi cient of static friction is ms 0.500, determine the smallest angle the ladder can make with the fl oor without slipping. 72. Two astronauts (Fig. P8.72), each having a mass of 75.0 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.00 m/s. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum and (b) the rotational energy of the system. By pulling on the rope, the astronauts shorten the distance between them to 5.00 m. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new rotational energy of the system? (f) How much work is done by the astronauts in shortening the rope?

Two astronauts (Fig. P8.72), each having a mass of 75.0 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.00 m/s. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum and (b) the rotational energy of the system. By pulling on the rope, the astronauts shorten the distance between them to 5.00 m. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new rotational energy of the system? (f) How much work is done by the astronauts in shortening the rope?

Two astronauts (Fig. P8.72), each having a mass M, are connected by a rope of length d having negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed v. (a) Calculate the magnitude of the angular momentum of the system by treating the astronauts as particles. (b) Calculate the rotational energy of the system. By pulling on the rope, the astronauts shorten the distance between them to d/2. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new rotational energy of the system? (f) How much work is done by the astronauts in shortening the rope?

Two window washers, Bob and Joe, are on a 3.00-m-long, 345-N scaffold supported by two cables attached to its ends. Bob weighs 750 N and stands 1.00 m from the left end, as shown in Figure P8.74. Two meters from the left end is the 500-N washing equipment. Joe is 0.500 m from the right end and weighs 1 000 N. Given that the scaffold is in rotational and translational equilibrium, what are the forces on each cable?

A star with mass 3.00 1030 kg and radius 1.50 109 m rotates on its axis at a rate of 0.010 0 rev/d. If the star suddenly collapses to a neutron star of radius 15.0 km, fi nd (a) the angular speed of the star and (b) the tangential speed of an indestructible astronaut standing on the equator.

A light rod of length 2L is free to rotate in a vertical plane about a frictionless pivot through its center. A particle of mass m1 is attached at one end of the rod, and a mass m2 is at the opposite end, where m1

m2. The system is released from rest in the vertical position shown in Figure P8.76a, and at some later time the system is rotating in the position shown in Figure P8.76b. Take the reference point of the gravitational potential energy to be at the pivot. (a) Find an expression for the systems total mechanical energy in the vertical position. (b) Find an expression for the total mechanical energy in the rotated position shown in Figure P8.76b. (c) Using the fact that the mechanical energy of the system is conserved, how would you determine the angular speed v of the system in the rotated position? (d) Find the magnitude of the torque on the system in the vertical position and in the rotated position. Is the torque constant? Explain what these results imply regarding the angular momentum of the system. (e) Find an expression for the magnitude of the angular acceleration of the system in the rotated position. Does your result make sense when the rod is horizontal? When it is vertical? Explain.

In Figure P8.77, the sliding block has a mass of 0.850 kg, the counterweight has a mass of 0.420 kg, and the pulley is a uniform solid cylinder with a mass of 0.350 kg and an outer radius of 0.030 0 m. The coeffi cient of kinetic friction between the block and the horizontal surface is 0.250. The pulley turns without friction on its axle. The light cord does not stretch and does not slip on the pulley. The block has a velocity of 0.820 m/s toward the pulley when it passes through a photogate. (a) Use energy methods to predict the speed of the block after it has moved to a second photogate 0.700 m away. (b) Find the angular speed of the pulley at the same moment.

(a) Without the wheels, a bicycle frame has a mass of 8.44 kg. Each of the wheels can be roughly modeled as a uniform solid disk with a mass of 0.820 kg and a radius of 0.343 m. Find the kinetic energy of the whole bicycle when it is moving forward at 3.35 m/s. (b) Before the invention of a wheel turning on an axle, ancient people moved heavy loads by placing rollers under them. (Modern people use rollers, too: Any hardware store will sell you a roller bearing for a lazy Susan.) A stone block of mass 844 kg moves forward at 0.335 m/s, supported by two uniform cylindrical tree trunks, each of mass 82.0 kg and radius 0.343 m. There is no slipping between the block and the rollers or between the rollers and the ground. Find the total kinetic energy of the moving objects.

In exercise physiology studies, it is sometimes important to determine the location of a persons center of gravity. This can be done with the arrangement shown in Figure P8.79. A light plank rests on two scales that read Fg1 380 N and Fg2 320 N. The scales are separated by a distance of 2.00 m. How far from the womans feet is her center of gravity?

In a circus performance, a large 5.0-kg hoop of radius 3.0 m rolls without slipping. If the hoop is given an angular speed of 3.0 rad/s while rolling on the horizontal ground and is then allowed to roll up a ramp inclined at 20 with the horizontal, how far along the incline does the hoop roll?

A uniform solid cylinder of mass M and radius R rotates on a frictionless horizontal axle (Fig. P8.81). Two objects with equal masses m hang from light cords wrapped around the cylinder. If the system is released from rest, fi nd (a) the tension in each cord and (b) the acceleration of each object after the objects have descended a distance h.

A painter climbs a ladder leaning against a smooth wall. At a certain height, the ladder is on the verge of slipping. (a) Explain why the force exerted by the vertical wall on the ladder is horizontal. (b) If the ladder of length L leans at an angle u with the horizontal, what is the lever arm for this horizontal force with the axis of rotation taken at the base of the ladder? (c) If the ladder is uniform, what is the lever arm for the force of gravity acting on the ladder? (d) Let the mass of the painter be 80 kg, L 4.0 m, the ladders mass be 30 kg, u 53, and the coeffi cient of friction between ground and ladder be 0.45. Find the maximum distance the painter can climb up the ladder. 83. A war-wolf, or trebuchet, is a device used during the Middle Ages to throw rocks at castles and now sometimes used to fl ing pumpkins and pianos. A simple trebuchet is shown in Figure P8.83. Model it as a stiff rod of negligible mass 3.00 m long and joining particles of mass 60.0 kg and 0.120 kg at its ends. It can turn on a frictionless horizontal axle perpendicular to the rod and 14.0 cm from the particle of larger mass. The rod is released from rest in a horizontal orientation. Find the maximum speed that the object of smaller mass attains.

A string is wrapped around a uniform cylinder of mass M and radius R. The cylinder is released from rest with the string vertical and its top end tied to a fi xed bar (Fig. P8.84). Show that (a) the tension in the string is one-third the weight of the cylinder, (b) the magnitude of the acceleration of the center of gravity is 2g/3, and (c) the speed of the center of gravity is (4gh/3)1/2 after the cylinder has descended through distance h. Verify your answer to part (c) with the energy approach.

The Iron Cross When a gymnast weighing 750 N executes the iron cross as in Figure P8.85a, the primary muscles involved in supporting this position are the latissimus dorsi (lats) and the pectoralis major (pecs). The rings exert an upward force on the arms and support the weight of the gymnast. The force exerted by the shoulder joint on the arm is labeled F S s while the two muscles exert a total force F S m on the arm. Estimate the magnitude of the force F S m. Note that one ring supports half the weight of the gymnast, which is 375 N as indicated in Figure P8.85b. Assume that the force F S m acts at an angle of 45 below the horizontal at a distance of 4.0 cm from the shoulder joint. In your estimate, take the distance from the shoulder joint to the hand to be 70 cm and ignore the weight of the arm.

Swinging on a high bar The gymnast shown in Figure P8.86 is performing a backwards giant swing on the high bar. Starting from rest in a near-vertical orientation, he rotates around the bar in a counterclockwise direction, keeping his body and arms straight. Friction between the bar and the gymnasts hands exerts a constant torque opposing the rotational motion. If the angular velocity of the gymnast at position 2 is measured to be 4.0 rad/s, determine his angular velocity at position 3. (Note that this maneuver is called a backwards giant swing, even though the motion of the gymnast would seem to be forwards.)

A 4.00-kg mass is connected by a light cord to a 3.00-kg mass on a smooth surface (Fig. P8.87). The pulley rotates about a frictionless axle and has a moment of inertia of 0.500 kg m2 and a radius of 0.300 m. Assuming that the cord does not slip on the pulley, fi nd (a) the acceleration of the two masses and (b) the tensions T1 and T2.

A 10.0-kg monkey climbs a uniform ladder with weight w 1.20 102 N and length L 3.00 m as shown in Figure P8.88. The ladder rests against the wall at an angle of u 60.0. The upper and lower ends of the ladder rest on frictionless surfaces, with the lower end fastened to the wall by a horizontal rope that is frayed and that can support a maximum tension of only 80.0 N. (a) Draw a freebody diagram for the ladder. (b) Find the normal force exerted by the bottom of the ladder. (c) Find the tension in the rope when the monkey is two-thirds of the way up the ladder. (d) Find the maximum distance d that the monkey can climb up the ladder before the rope breaks. (e) If the horizontal surface were rough and the rope were removed, how would your analysis of the problem be changed and what other information would you need to answer parts (c) and (d)? FIG