Solved: How much work is done by the gravitational force

Problem 49GP A ?conical pendulum is formed by attaching a 500 g ball to a 1.0-m-long string, then allowing the mass to move in a horizontal circle of radius 20 cm. Figure 49 shows that the string traces out the surface of a cone, hence the name. a. What is the tension in the string? b. What is the ball’s angular velocity, in rpm? Hint: Determine the horizontal and vertical components of the forces acting on the ball, and use the fact that the vertical component of acceleration is zero since there is no vertical motion. FIGURE 49

Problem 42GP In the Bohr model of the hydrogen atom, an electron orbits a proton at a distance of . The proton pulls on the electron with an electric force of . How many revolutions per second does the electron make?

Problem 41GP The car in Figure P6.51 travels at a constant speed along the road shown. Draw vectors showing its acceleration at the three points A, B, and C, or write . The lengths of your vectors should correspond to the magnitudes of the accelerations.

Problem 40GP How fast must a plane fly along the earth’s equator so that the sun stands still relative to the passengers? In which direction must the plane fly, east to west or west to east? Give your answer in both km/h and mph. The radius of the earth is 6400 km.

Problem 39P A vertical spring (ignore its mass), whose spring stiffness constant is 950 N/m, is attached to a table and is compressed down 0.150 m. (a) What upward speed can it give to a 0.30-kg ball when released? (b) How high above its original position (spring compressed) will the ball fly?

Problem 38P A projectile is fired at an upward angle of 45.0° from the top of a 265-m cliff with a speed of 185 m/s. What will be its speed when it strikes the ground below? (Use conservation of energy.)

(II) A trampoline artist jumps vertically upward from the top of a platform with a speed of . (a) How fast is he going as he lands on the trampoline, below (Fig. )? (b) If the trampoline behaves like a spring with spring stiffness constant \(6.2 \times 10^{4} \mathrm{~N} / \mathrm{m}\) how far does he depress it? Equation Transcription: Text Transcription: 6.2 x 10^4N/m

Problem 36P In the high jump, Fran’s kinetic energy is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must Fran leave the ground in order to lift her center of mass 2.10 m and cross the bar with a speed of 0.70 m/s?

(I) A sled is initially given a shove up a frictionless \(28.0^{\circ}\) incline. It reaches a maximum vertical height \(1.35 \mathrm{~m}\) higher than where it started. What was its initial speed?

Problem 34P A novice skier, starting from rest, slides down a frictionless 35.0° incline whose vertical height is 185 m. How fast is she going when she reaches the bottom?

Problem 33P Jane, looking for Tarzan, is running at top speed (5.3 m/s) and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does the length of the vine affect your answer?

Problem 33MCQ Questions concern a classic figure-skating jump called the axle. A skater starts the jump moving forward as shown in Figure, leaps into the air, and turns one-and-a-half revolutions before landing. The typical skater is in the air for about 0.5 s, and the skater’s hands are located about 0.8 m from the rotation axis. FIGURE 31 What is the approximate speed of the skater’s hand? A. 1 m/s B. 3 m/s C. 9 m/s D. 15 m/s

Problem 32P A spring with k = 53 N/m hangs vertically next to a ruler. The end of the spring is next to the 15-cm mark on the ruler. If a 2.5-kg mass is now attached to the end of the spring, where will the end of the spring line up with the ruler marks?

Problem 28MCQ Currently, the moon goes around the earth once every 27.3 days. If the moon could be brought into a new circular orbit with a smaller radius, its orbital period would be A. More than 27.3 days. B. 27.3 days. C. Less than 27.3 days.

Problem 27P A 7.0-kg monkey swings from one branch to another 1.2 m higher. What is the change in potential energy?

Problem 48GP A 5.0 g coin is placed 15 cm from the center of a turntable. The coin has static and kinetic coefficients of friction with the turntable surface of . The turntable very slowly speeds up to 60 rpm. Does the coin slide off?

Problem 72GP Designers of today’s cars have built “5mi/h (8 km/h) bumpers” that are designed to compress and rebound elastically without any physical damage at speeds below 8 km/h. If the material of the bumpers permanently deforms after a compression of 1.5 cm, but remains like an elastic spring up to that point, what must the effective spring stiffness constant of the bumper be, assuming the car has a mass of 1300 kg and is tested by ramming into a solid wall?

Problem 73GP In a certain library the first shelf is 10.0 cm off the ground, and the remaining four shelves are each spaced 30.0 cm above the previous one. If the average book has a mass of 1.5 kg with a height of 21 cm, and an average shelf holds 25 books, how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start?

Problem 31P A 55-kg hiker starts at an elevation of 1600 m and climbs to the top of a 3300-m peak. (a) What is the hiker’s change in potential energy? (b) What is the minimum work required of the hiker? (c) Can the actual work done be more than this? Explain why.

Problem 32MCQ Questions concern a classic figure-skating jump called the axel. A skater starts the jump moving forward as shown in Figure, leaps into the air, and turns one-and-a-half revolutions before landing. The typical skater is in the air for about 0.5 s, and the skater’s hands are located about 0.8 m from the rotation axis. FIGURE 31 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 m/s2

Problem 43GP A 75 kg man weighs himself at the north pole and at the equator. Which scale reading is higher? By how much? Assume the earth is a perfect sphere. Explain why the readings differ.

Problem 44GP A 1500 kg car takes a 50-m-radius unbanked curve at 15 m/s. What is the size of the friction force on the car?

Problem 46GP Suppose the moon were held in its orbit not by gravity but by a massless cable attached to the center of the earth. What would be the tension in the cable? See the inside of the back cover for astronomical data.

Problem 47GP A 30 g ball rolls around a 40-cm-diameter L-shaped track, shown in Figure 47, at 60 rpm. Rolling friction can be neglected. a. How many different contact forces does the track exert on the ball? Name them. b. What is the magnitude of the net force on the ball? FIGURE 47

Problem 52GP While at the county fair, you decide to ride the Ferris wheel. Having eaten too many candy apples and elephant ears, you find the motion somewhat unpleasant. To take your mind off your stomach, you wonder about the motion of the ride. You estimate the radius of the big wheel to be 15 m, and you use your watch to find that each loop around takes 25 s. a. What are your speed and magnitude of your acceleration? b. What is the ratio of your apparent weight to your true weight at the top of the ride? c. What is the ratio of your apparent weight to your true weight at the bottom?

Problem 53GP A car drives over the top of a hill that has a radius of 50 m. What maximum speed can the car have without flying off the road at the top of the hill?

Problem 1CQ There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?

Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions—it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?

In what ways is the word "work" as used in everyday language the same as that defined in physics? In what ways is it different? Give examples of both.

Problem 2CQ Can centripetal acceleration change the speed of circular motion? Explain.

Problem 2PE Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?

Problem 2Q Can a centripetal force ever do work on an object? Explain.

Problem 3CQ If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you use large- or small diameter tires? Explain.

Problem 3PE An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?

Problem 3Q Can the normal force on an object ever do work? Explain.

Problem 4CQ Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force?

Problem 4PE (a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? 6? (c) Given that Earth has a radius of 6.4×10? m at its equator, what is the linear velocity at Earth’s surface?

Problem 5CQ If centripetal force is directed toward the center, why do you feel that you are ‘thrown’ away from the center as a car goes around a curve? Explain.

Problem 5PE A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher’s hand is 35.0 m/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?

Problem 6CQ Race car drivers routinely cut corners as shown in Figure 6.32. Explain how this allows the curve to be taken at the greatest speed.

Problem 6PE In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?

Problem 6Q Why is it tiring to push hard against a solid wall even though you are doing no work?

Problem 7CQ A number of amusement parks have rides that make vertical loops like the one shown in Figure 6.33. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if: (a) The car goes over the top at faster than this speed? (b)The car goes over the top at slower than this speed?

(II) A lever such as that shown in Fig. can be used to lift objects we might not otherwise be able to lift. Show that the ratio of output force, \(F_{0\), to input force,\(F_{1}\), is related to the lengths \(l_{1} \text { and } l_{O}\) from the pivot point by \(F_{O} / F_{I}=l_{I} / l_{O}\) (ignoring friction and the mass of the lever), given that the work output equals work input. Equation Transcription: Text Transcription: F0 F1 l1 and lO FO/FI=lI/lO

Problem 7Q You have two springs that are identical except that spring 1 is stiffer than spring 2(k1 > K2). On which spring is more work done: (a) if they are stretched using the same force; (b) if they are stretched the same distance?

Problem 8CQ What is the direction of the force exerted by the car on the passenger as the car goes over the top of the amusement ride pictured in Figure 6.33 under the following circumstances: (a) The car goes over the top at such a speed that the gravitational force is the only force acting? (b) The car goes over the top faster than this speed? (c) The car goes over the top slower than this speed?

Problem 8PE Integrated Concepts When kicking a football, the kicker rotates his leg about the hip joint. (a) If the velocity of the tip of the kicker’s shoe is 35.0 m/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip’s angular velocity? (b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m/s? (c) Find the maximum range of the football, neglecting air resistance.

A hand exerts a constant horizontal force on a block that is free to slide on a frictionless surface (Fig. ). The block starts from rest at point , and by the time it has traveled a distance to point it is traveling with speed \(v_{B}\). When the block has traveled another distance to point will its speed be greater than, less than, or equal to \(2 v_{B}\) ? Explain your reasoning. Equation Transcription: Text Transcription: v_B 2_vB

Problem 9CQ As a skater forms a circle, what force is responsible for making her turn? Use a free body diagram in your answer.

Problem 9PE Construct Your Own Problem Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders?’

Problem 10PE A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity?

In Fig. , water balloons are tossed from the roof of a building, all with the same speed but with different launch angles. Which one has the highest speed on impact? Ignore air resistance.

Problem 11CQ Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car’s speed? What is the direction of the force exerted on you by the car seat?

(II) The force on an object, acting along the x axis, varies as shown in Fig. 6-37. Determine the work done by this force to move the object (a) from x=0.0 to \(x=10.0 \mathrm{~m}\), and (b) from x=0.0 to \(x=15.0 \mathrm{~m}\).

Problem 12CQ Suppose a mass is moving in a circular path on a frictionless table as shown in figure. In the Earth’s frame of reference, there is no centrifugal force pulling the mass away from the center of rotation, yet there is a very real force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and Newton’s third law, explain what force stretches the string, identifying its physical origin.

Problem 15P (I) At room temperature, an oxygen molecule, with mass of 5.31 X 10-26 kg typically has a kinetic energy of about 6.21 X 10-26 kg, How fast is it moving?

Problem 19P An 88-g arrow is fired from a bow whose string exerts an average force of 110 N on the arrow over a distance of 78 cm. What is the speed of the arrow as it leaves the bow?

Describe the energy transformations when a child hops around on a pogo stick.

Problem 19Q Two identical arrows, one with twice the speed of the other, are fired into a bale of hay. Assuming the hay exerts a constant "frictional" force on the arrows, the faster arrow will penetrate how much farther than the slower arrow? Explain.

Problem 14Q What happens to the gravitational potential energy when water at the top of a waterfall falls to the pool below?

Problem 20Q Analyze the motion of a simple swinging pendulum in terms of energy, (a) ignoring friction, and (b) taking friction into account. Explain why a grandfather clock has to be wound up.

Problem 22Q Suppose you lift a suitcase from the floor to a table. The work you do on the suitcase depends on which of the following: (a) whether you lift it straight up or along a more complicated path, (b) the time the lifting takes, (c) the height of the table, and (d) the weight of the suitcase? Step-by-step solution

Problem 23Q Repeat Question 21 for the power needed instead of the work. 21. Suppose you lift a suitcase from the floor to a table. The work you do on the suitcase depends on which of the following: (a) whether you lift it straight up or along a more complicated path, (b) the time the lifting takes, (c) the height of the table, and (d) the weight of the suitcase?

Problem 24Q Why is it easier to climb a mountain via a zigzag trail rather than to climb straight up?

Recall from Chapter 4, Example , that you can use a pulley and ropes to decrease the force needed to raise a heavy load (see Fig. ). But for every meter the load is raised, how much rope must be pulled up? Account for this, using energy concepts.

(III) An engineer is designing a spring to be placed at the bottom of an elevator shaft. If the elevator cable should break when the elevator is at a height h above the top of the spring, calculate the value that the spring stiffness constant k should have so that passengers undergo an acceleration of no more than \(5.0 \mathrm{~g}\) when brought to rest. Let M be the total mass of the elevator and passengers.

Problem 61P (I) (a) Show that one British horsepower (550 ft -lb/s) is equal to 746 W. (6) What is the horsepower rating of a 75-Wlightbulb?

Problem 65P (II) A shot-putter accelerates a 7.3-kg shot from rest to 14 m/s in 1.5 s. What average power was developed?

A ball is attached to a horizontal cord of length whose other end is fixed (Fig. ). If the ball is released, what will be its speed at the lowest point of its path? (b) A peg is located a distance directly below the point of attachment of the cord. If \(h=0.80 L\) what will be the speed of the ball when it reaches the top of its circular path about the peg? Equation Transcription: Text Transcription: h=0.80 L

Problem 16Q Describe the energy transformations when a child hops around on a pogo stick (there is a spring inside).

Problem 1P How much work is done by the gravitational force when a 265-kg pile driver falls 2.80 m?

Problem 2P A 65.0-kg firefighter climbs a flight of stairs 20.0 m high. How much work is required?

Problem 3P A 1300-N crate rests on the floor. How much work is required to move it at constant speed (a) 4.0 m along the floor against a friction force of 230 N, and (b) 4.0 m vertically?

(I) How much work did the movers do (horizontally) pushing a \(160-\mathrm{kg}\) crate \(10.3 \mathrm{~m}\) across a rough floor without acceleration, if the effective coefficient of friction was 0.50?

Problem 4Q A woman swimming upstream is not moving with respect to the shore. Is she doing any work? If she stops swimming and merely floats, is work done on her?

(II) A box of mass 5.0 kg is accelerated from rest across a floor at a rate of \(2.0 \ \mathrm {m/s}^2\) for 7.0 s. Find the net work done on the box.

Problem 5Q Is the work done by kinetic friction forces always negative? [Hint: Consider what happens to the dishes when you pull a tablecloth out from under them.]

Problem 6P Eight books, each 4.3 cm thick with mass 1.7 kg, lie flat on a table. How much work is required to stack them one on top of another?

A \(330-\mathrm{kg}\) piano slides \(3.6 \mathrm{~m}\) down a \(28^{\circ}\) incline and is kept from accelerating by a man who is pushing back on it parallel to the incline (Fig. 6-36). The effective coefficient of kinetic friction is 0.40. Calculate: (a) the force exerted by the man, (b) the work done by the man on the piano, (c) the work done by the friction force, (d) the work done by the force of gravity, and (e) the net work done on the piano.

Problem 9P (a) Find the force required to give a helicopter of mass M an acceleration of 0.10 g upward. (b) Find the work done by this force as the helicopter moves a distance h upward.

Problem 9Q By approximately how much does your gravitational potential energy change when you jump as high as you can?

(II) What is the minimum work needed to push a \(950-\mathrm{kg}\) car \(810 \mathrm{~m}\) up along a \(9.0^{\circ}\) incline? (a) Ignore friction. (b) Assume the effective coefficient of friction retarding the car is 0.25.

(II) In Fig. 6-6a, assume the distance axis is linear and that \(d_{\mathrm{A}}=10.0 \mathrm{~m}\) and \(d_{\mathrm{B}}=35.0 \mathrm{~m}\). Estimate the work done by force F in moving a 2.80-kg object from \(d_{\mathrm{A}}\) to \(d_{\mathrm{B}}\).

A pendulum is launched from a point that is a height above its lowest point in two different ways (Fig. 6-32). During both launches, the pendulum is given an initial speed of On the first launch, the initial velocity of the pendulum is directed upward along the trajectory, and on the second launch it is directed downward along the trajectory. Which launch will cause it to swing the largest angle from the equilibrium position? Explain.

Problem 12Q A coil spring of mass m rests upright on a table. If you compress the spring by pressing down with your hand and then release it, can the spring leave the table? Explain, using the law of conservation of energy.

Problem 13P A spring has k = 88 N/m. Use a graph to determine the work needed to stretch it from x = 3.8 cm to x = 5.8 cm, where x is the displacement from its unstretched length.

A bowling ball is hung from the ceiling by a steel wire (Fig. 6-33). The instructor pulls the ball back and stands against the wall with the ball against his nose. To avoid injury the instructor is supposed to release the ball without pushing it. Why?

(II) The net force exerted on a particle acts in the +x direction. Its magnitude increases linearly from zero at x=0, to \(24.0 \mathrm{~N}\) at \(x=3.0 \mathrm{~m}\). It remains constant at \(24.0 \mathrm{~N}\) from \(x=3.0 \mathrm{~m}\) to \(x=8.0 \mathrm{~m}\), and then decreases linearly to zero at \(x=13.0 \mathrm{~m}\). Determine the work done to move the particle from x=0 to \(x=13.0 \mathrm{~m}\) graphically by determining the area under the \(F_x\) vs x graph.

Problem 16P (a) If the KE of an arrow is doubled, by what factor has its speed increased? (b) If its speed is doubled, by what factor does its KE increase?

Problem 17P How much work is required to stop an electron (m = 9.11 × 10?31 kg) which is moving with a speed of 1.90 × 106 m/s?

Problem 17Q A child on a sled (total mass m) starts from rest at the top of a hill of height h and slides down. Does the velocity at the bottom depend on the angle of the hill if (a) it is icy and there is no friction, and (b) there is friction (deep snow)?

Seasoned hikers prefer to step over a fallen log in their path rather than stepping on top and jumping down on the other side. Explain.

Problem 18P How much work must be done to stop a 1250-kg car traveling at 105 km/h?

Problem 20P A baseball (m = 140 g) traveling 32 m/s moves a fielder’s glove backward 25 cm when the ball is caught. What was the average force exerted by the ball on the glove?

(II) If the speed of a car is increased by 50%, by what factor will its minimum braking distance be increased, assuming all else is the same? Ignore the driver’s reaction time.

When a “superball” is dropped, can it rebound to a height greater than its original height? Explain.

Problem 22P At an accident scene on a level road, investigators measure a car’s skid mark to be 88 m long. The accident occurred on a rainy day, and the coefficient of kinetic friction was estimated to be 0.42. Use these data to determine the speed of the car when the driver slammed on (and locked) the brakes. (Why does the car’s mass not matter?)

(II) A softball having a mass of 0.25 kg is pitched at 95 km/h. By the time it reaches the plate, it may have slowed by 10%. Neglecting gravity, estimate the average force of air resistance during a pitch, if the distance between the plate and the pitcher is about 15 m.

(III) A \(285-\mathrm{kg}\) load is lifted \(22.0 \mathrm{~m}\) vertically with an acceleration \(a=0.160 \mathrm{~g}\) by a single cable. Determine (a) the tension in the cable, (b) the net work done on the load. (c) the work done by the cable on the load, (d) the work done by gravity on the load, and (e) the final speed of the load assuming it started from rest.

Problem 24P How high will a 1.85-kg rock go if thrown straight up by someone who does 80.0 J of work on it? Neglect air resistance.

A film of Jesse Owens's famous long jump (Fig. ) in the 1936 Olympics shows that his center of mass rose from launch point to the top of the arc. What minimum speed did he need at launch if he was traveling at at the top of the arc?

(I) A spring has a spring stiffness constant, k, of 440 N/m. How much must this spring be stretched to store 25 J of potential energy?

(I) By how much does the gravitational potential energy of a \(64-\mathrm{kg}\) pole vaulter change if his center of mass rises about \(4.0 \mathrm{~m}\) during the jump?

(II) A \(1200-\mathrm{kg}\) car rolling on a horizontal surface has speed \(v=65 \mathrm{~km} / \mathrm{h}\) when it strikes a horizontal coiled spring and is brought to rest in a distance of \(2.2 \mathrm{~m}\). What is the spring stiffness constant of the spring?

(II) A \(1.60-\mathrm{m}\) tall person lifts a \(2.10-\mathrm{kg}\) book from the ground so it is \(2.20 \mathrm{~m}\) above the ground. What is the potential energy of the book relative to (a) the ground, and (b) the top of the person's head? (c) How is the work done by the person related to the answers in parts (a) and (b) ?

(II) A block of mass slides without friction along the looped track shown in Fig. . If the block is to remain on the track, even at the top of the circle (whose radius is ), from what minimum height must it be released?

(II) A 62-kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 12 m, and falls a total of 31 m. (a) Calculate the spring stiffness constant k of the bungee cord, assuming Hooke’s law applies. (b) Calculate the maximum acceleration she experiences.

(II) A block of mass is attached to the end of a spring (spring stiffness constant ), Fig. . The block is given an initial displacement \(x_{0}\), after which it oscillates back and forth. Write a formula for the total mechanical energy (ignore friction and the mass of the spring) in terms of \(x_{0}\), position , and speed . Equation Transcription: Text Transcription: X_0 x_0

(II) The roller-coaster car shown in Fig. is dragged up to point 1 where it is released from rest. Assuming no friction, calculate the speed at points 2,3, and

Problem 44P A 0.40-kg ball is thrown with a speed of 12 m/s at an angle of 33°. (a) What is its speed at its highest point, and (b) how high does it go? (Use conservation of energy, and ignore air resistance.)

Problem 46P A cyclist intends to cycle up a 7.8° hill whose vertical height is 150 m. Assuming the mass of bicycle plus cyclist is 75 kg, (a) calculate how much work must be done against gravity. (b) If each complete revolution of the pedals moves the bike 5.1 m along its path, calculate the average force that must be exerted on the pedals tangent to their circular path. Neglect work done by friction and other losses. The pedals turn in a circle of diameter 36 cm.

(I) Two railroad cars, each of mass \(7650 \mathrm{~kg}\) and traveling \(95 \mathrm{~km} / \mathrm{h}\) in opposite directions, collide head-on and come to rest. How much thermal energy is produced in this collision?

Problem 48P A 21.7-kg child descends a slide 3.5 m high and reaches the bottom with a speed of 2.2 m/s. How much thermal energy due to friction was generated in this process?

Problem 49P A ski starts from rest and slides down a 22° incline 75 m long. (a) If the coefficient of friction is 0.090, what is the ski’s speed at the base of the incline? (b) If the snow is level at the foot of the incline and has the same coefficient of friction, how far will the ski travel along the level? Use energy methods.

Problem 50P A 145-g baseball is dropped from a tree 13.0 m above the ground. (a) With what speed would it hit the ground if air resistance could be ignored? (b) If it actually hits the ground with a speed of 8.00 m/s, what is the average force of air resistance exerted on it?

Problem 52P A 110-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 350 N. For the first 15 m the floor is frictionless, and for the next 15 m the coefficient of friction is 0.30. What is the final speed of the crate?

Problem 51P You drop a ball from a height of 2.0 m, and it bounces back to a height of 1.5 m. (a) What fraction of its initial energy is lost during the bounce? (b) What is the ball’s speed just as it leaves the ground after the bounce? (c) Where did the energy go?

(II) Suppose the roller coaster in Fig. passes point 1 with a speed of \(1.70 \mathrm{~m} / \mathrm{s}\). If the average force of friction is equal to one-fifth of its weight, with what speed will it reach point 2 ? The distance traveled is . Equation Transcription: Text Transcription: 1.70 m/s

(II) A skier traveling \(12.0 \mathrm{~m} / \mathrm{s}\) reaches the foot of a steady upward \(18.0^{\circ}\) incline and glides \(12.2 \mathrm{~m}\) up along this slope before coming to rest. What was the average coefficient of friction?

(III) A wood block is firmly attached to a very light horizontal spring \((k=180 \mathrm{~N} / \mathrm{m})\) as shown in Fig. It is noted that the block-spring system, when compressed and released, stretches out beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table? Equation Transcription: Text Transcription: (k=180 N/m)

A 280-g wood block is firmly attached to a very light horizontal spring. Fig. 6-40. The block can slide along a table where the coefficient of friction is 0.30. A force of \(22 \mathrm{~N}\) compresses the spring \(18 \mathrm{~cm}\). If the spring is released from this position, how far beyond its equilibrium position will it stretch at its first maximum extension?

(III) Early test flights for the space shuttle used a "glider" (mass of \(980 \mathrm{~kg}\) including pilot) that was launched horizontally at \(500 \mathrm{~km} / \mathrm{h}\) from a height of \(3500 \mathrm{~m}\). The glider eventually landed at a speed of \(200 \mathrm{~km} / \mathrm{h}\). (a) What would its landing speed have been in the absence of air resistance? (b) What was the average force of air resistance exerted on it if it came in at a constant glide of \(10^{\circ}\) to the Earth?

(I) How long will it take a 1750-W motor to lift a 315-kg piano to a sixth-story window 16.0 m above?

Problem 59P If a car generates 18 hp when traveling at a steady 88 km/h, what must be the average force exerted on the car due to friction and air resistance?

Problem 60P A 1400-kg sports car accelerates from rest to 95 km/h in 7.4 s. What is the average power delivered by the engine?

Problem 62P Electric energy units are often expressed in the form of “kilowatt-hours.” (a) Show that one kilowatt-hour (kWh) is equal to 3.6 × 106J. (b) If a typical family of four uses electric energy at an average rate of 520 W, how many kWh would their electric bill be for one month, and (c) how many joules would this be? (d) At a cost of $0.12 per kWh, what would their monthly bill be in dollars? Does the monthly bill depend on the rate at which they use the electric energy?

(II) A driver notices that her 1150-kg car slows down from 85 km/h to 65 km/h in about 6.0 s on the level when it is in neutral. Approximately what power (watts and hp) is needed to keep the car traveling at a constant 75 km/h?

(II) How much work can a 3.0-hp motor do in 1.0 h?

A pump is to lift 18.0 kg of water per minute through a height of 3.60 m. What output rating (watts) should the pump motor have?

Problem 67P During a workout, the football players at State U. ran up the stadium stairs in 66 s. The stairs are 140 m long and inclined at an angle of 32°. If a typical player has a mass of 95 kg, estimate the average power output on the way up. Ignore friction and air resistance.

Problem 68P How fast must a cyclist climb a 6.0° hill to maintain a power output of 0.25 hp? Neglect work done by friction, and assume the mass of cyclist plus bicycle is 68 kg.

Problem 69P A 1200-kg car has a maximum power output of 120 hp. How steep a hill can it climb at a constant speed of 75 km/h if the frictional forces add up to 650 N?

Problem 70P What minimum horsepower must a motor have to be able to drag a 310-kg box along a level floor at a speed of 1.20 m/s if the coefficient of friction is 0.45?

Problem 71P A bicyclist coasts down a 7.0° hill at a steady speed of 5.0 m/s. Assuming a total mass of 75 kg (bicycle plus rider), what must be the cyclist’s power output to climb the same hill at the same speed?

The block of mass m sliding without friction along the looped track shown in Fig. 6-39 is to remain on the track at all times, even at the very top of the loop of radius r. (a) In terms of the given quantities, determine the minimum release height h (as in Problem 40). Next, if the actual release height is 2h, calculate (b) the normal force exerted by the track at the bottom of the loop, (c) the normal force exerted by the track at the top of the loop, and (d) the normal force exerted by the track after the block exits the loop onto the flat section.

Problem 76GP An airplane pilot fell 370 m after jumping from an aircraft without his parachute opening. He landed in a snowbank, creating a crater 1.1m deep, but survived with only minor injuries. Assuming the pilot’s mass was 78 kg and his terminal velocity was 35 m/s, estimate (a) the work done by the snow in bringing him to rest; (b) the average force exerted on him by the snow to stop him; and (c) the work done on him by air resistance as he fell.

Problem 78GP A 65-kg hiker climbs to the top of a 3700-m-high mountain. The climb is made in 5.0 h starting at an elevation of 2300 m. Calculate (a) the work done by the hiker against gravity, (b) the average power output in watts and in horsepower, and (c) assuming the body is 15% efficient, what rate of energy input was required.

Problem 79GP An elevator cable breaks when a 920-kg elevator is 28 m above a huge spring (k = 2.2 × 105N/m) at the bottom of the shaft. Calculate (a) the work done by gravity on the elevator before it hits the spring, (b) the speed of the elevator just before striking the spring, and (c) the amount the spring compresses (note that work is done by both the spring and gravity in this part).

Squaw Valley ski area in California claims that its lifts can move 47,000 people per hour. If the average lift carries people about 200 m (vertically) higher, estimate the power needed.

Problem 81GP Water flows (v ? 0) over a dam at the rate of 650 kg/s and falls vertically 81 m before striking the turbine blades. Calculate (a) the speed of the water just before striking the turbine blades (neglect air resistance), and (b) the rate at which mechanical energy is transferred to the turbine blades, assuming 58% efficiency.

Show that on a roller coaster with a circular vertical loop (Fig. 6-44), the difference in your apparent weight at the top of the circular loop and the bottom of the circular loop is 6 g’s-that is, six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this answer doesn't depend on the size of the loop or how fast you go through it.

Problem 83GP (a) If the human body could convert a candy bar directly into work, how high could an 82-kg man climb a ladder if he were fueled by one bar (= 1100kJ)? (b) If the man then jumped off the ladder, what will be his speed when he reaches the bottom?

Problem 84GP A projectile is fired at an upward angle of 45.0° from the top of a 165-m cliff with a speed of 175 m/s. What will be its speed when it strikes the ground below? (Use conservation of energy and neglect air resistance.)

If you stand on a bathroom scale, the spring inside the scale compresses \(0.60 \mathrm{~mm}\), and it tells you your weight is \(710 \mathrm{~N}\). Now if you jump on the scale from a height of \(1.0 \mathrm{~m}\), what does the scale read at its peak?

A student runs at \(5.0 \mathrm{~m} / \mathrm{s}\), grabs a rope, and swings out over a lake (Fig. ). He releases the rope when his velocity is zero. (a) What is the angle \(\theta\) when he releases the rope? (b) What is the tension in the rope just before he releases it? (c) What is the maximum tension in the rope? Equation Transcription: Text Transcription: 5.0 m/s \theta

In the rope climb, a 72-kg athlete climbs a vertical distance of 5.0 m in 9.0 s. What minimum power output was used to accomplish this feat?

Problem 88GP Some electric-power companies use water to store energy. Water is pumped by reversible turbine pumps from a low to a high reservoir. To store the energy produced in 1.0 hour by a 120-MW (120 × 106W) electric-power plant, how many cubic meters of water will have to be pumped from the lower to the upper reservoir? Assume the upper reservoir is 520 m above the lower and we can neglect the small change in depths within each. Water has a mass of 1000 kg for every 1.0 m3.

A spring with spring stiffness constant k is cut in half. What is the spring stiffness constant for each of the two resulting springs?

Problem 90GP A 6.0-kg block is pushed 8.0 m up a rough 37° inclined plane by a horizontal force of 75 N. If the initial speed of the block is 2.2 m/s up the plane and a constant kinetic friction force of 25 N opposes the motion, calculate (a) the initial kinetic energy of the block; (b) the work done by the 75-N force; (c) the work done by the friction force; (d) the work done by gravity; (e) the work done by the normal force; (f) the final kinetic energy of the block.

Problem 91GP If a 1500-kg car can accelerate from 35 km/h to 55 km/h in 3.2 s, how long will it take to accelerate from 55 km/h to 75 km/h? Assume the power stays the same, and neglect frictional losses.

In a common test for cardiac function (the "stress test"), the patient walks on an inclined treadmill (Fig. 6-46). Estimate the power required from a patient when the treadmill is sloping at an angle of \(15^{\circ}\) and the velocity is \(3.3 \mathrm{~km} / \mathrm{h}\). (How does this power compare to the power rating of a lightbulb?) Equation Transcription: Text Transcription: 15° 3.3 km/h

Problem 93GP (a) If a volcano spews a 500-kg rock vertically upward a distance of 500 m, what was its velocity when it left the volcano? (b) If the volcano spews the equivalent of 1000 rocks of this size every minute, what is its power output?

Water falls onto a water wheel from a height of \(2.0 \mathrm{~m}\) at a rate of \(95 \mathrm{~kg} / \mathrm{s}\). (a) If this water wheel is set up to provide electricity output, what is its maximum power output? (b) What is the speed of the water as it hits the wheel?