Solved: Consider the glider of Example 7.7 (Section 7.2)

Problem 87CP CALC? A proton with mass m moves in one dimension. The potential-energy function is U(x) = ?/x2) – (?/x), where ? and ? are positive constants. The proton is released from rest at x0 = ?/?. (a) Show that U(x) can be written as Graph U(x). Calculate U(x0) and thereby locate the point x 0 on the graph. (b) Calculate v(x), the speed of the proton as a function of position. Graph v(x) and give a qualitative description of the motion. (c) For what value of x is the speed of the proton a maximum? What is the value of that maximum speed? (d) What is the force on the proton at the point in part (c)? (e) Let the pro-ton be released instead at x1 = 3?/?. Locate the point x1 on the graph of U(x). Calculate v(x) and give a qualitative description of the motion. (f) For each release point (x = x0 and x = x1), what are the maximum and minimum values of x reached during the motion?

Problem 1DQ A baseball is thrown straight up with initial speed v0. If air resistance cannot be ignored, when the ball returns to its initial height its speed is less than v0. Explain why, using energy concepts.

Problem 2DQ A projectile has the same initial kinetic energy no matter what the angle of projection. Why doesn’t it rise to the same maximum height in each case?

Problem 2E BIO How High Can We Jump?? The maximum height a typical human can jump from a crouched start is about 60 cm. By how much does the gravitational potential energy increase for a 72-kg person in such a jump? Where does this energy come from?

Problem 1E In one day, a 75-kg mountain climber ascends from the 1500-m level on a vertical cliff to the top at 2400 m. The next day, she descends from the top to the base of the cliff, which is at an elevation of 1350 m. What is her change in gravitational potential energy (a) on the first day and (b) on the second day?

Problem 3DQ An object is released from rest at the top of a ramp. If the ramp is frictionless, does the object’s speed at the bottom of the ramp depend on the shape of the ramp or just on its height? Explain. What if the ramp is? ot? frictionless?

Problem 3E CP? A 90.0-kg mail bag hangs by a vertical rope 3.5 m long. A postal worker then displaces the bag to a position 2.0 m sideways from its original position, always keeping the rope taut. (a) What horizontal force is necessary to hold the bag in the new position? (b) As the bag is moved to this position, how much work is done (i) by the rope and (ii) by the worker?

Problem 4DQ An egg is released from the rest from the roof of a building and falls to the ground. Its fall is observed by a student on the roof of the building, who uses coordinates with origin at the roof, and by a student on the ground, who uses coordinates with origin at the ground. Do the values the two students assign to the following quantities match each other: initial gravitational potential energy, final gravitational potential energy, change in gravitational potential energy, and kinetic energy of the egg just before it strikes the ground? Explain.

Problem 4E BIO Food Calories. The ?food calorie,? equal to 4186 J, is a measure of how much energy is released when the body metabolizes food. A certain fruit-and-cereal bar contains 140 food calories. (a) If a 65-kg hiker eats one bar, how high a mountain must he climb to “work off” the calories, assuming that all the food energy goes into increasing gravitational potential energy? (b) If, as is typical, only 20% of the food calories go into mechanical energy, what would be the answer to part (a)? (?Note?: In this and all other problems, we are assuming that 100% of the food calories that are eaten are absorbed and used by the body. This is not true. A person’s “metabolic efficiency” is the percentage of calories eaten that are actually used; the body eliminates the rest. Metabolic efficiency varies considerably from person to person.)

Problem 5DQ A physics teacher had a bowling ball suspended from a very long rope attached to the high ceiling of a large lecture hall. To illustrate his faith in conservation of energy, he would back up to one side of the stage, pull the ball far to one side until the taut rope brought it just to the end of his nose, and then release it. The massive ball would swing in a mighty arc across the stage and then return to stop momentarily just in front of the nose of the stationary, unflinching teacher. However, one day after the demonstration he looked up just in time to see a student at the other side of the stage ?push? the ball away from his nose as he tried to duplicate the demonstration. Tell the rest of the story, and explain the reason for the potentially tragic outcome.

Problem 5E A baseball is thrown from the roof of a 22.0-m-tall building with an initial velocity of magnitude 12.0 m/s and directed at an angle of 53.1o above the horizontal. (a) What is the speed of the ball just before it strikes the ground? Use energy methods and ignore air resistance. (b) What is the answer for part (a) if the initial velocity is at an angle of 53.1o below the horizontal? (c) If the effects of air resistance are included, will part (a) or (b) give the higher speed?

Problem 6DQ Lost Energy? The principle of the conservation of energy tells us that energy is never lost, but only changes from one form to another. Yet in many ordinary situations, energy may appear to be lost. In each case, explain what happens to the “lost” energy. (a) A box sliding on the floor comes to a halt due to friction. How did friction take away its kinetic energy, and what happened to that energy? (b) A car stops when you apply the brakes. What happened to its kinetic energy? (c) Air resistance uses up some of the original gravitational potential energy of a falling object. What type of energy did the “lost” potential energy become? (d) When a returning space shuttle touches down on the run way, it has lost almost all its kinetic energy and gravitational potential energy. Where did all that energy go?

Problem 6E A crate of mass M starts from rest at the top of a frictionless ramp inclined at an angle ? above the horizontal. Find its speed at the bottom of the ramp, a distance d from where it started. Do this in two ways: Take the level at which the potential energy is zero to be (a) at the bottom of the ramp with y positive upward, and (b) at the top of the ramp with y positive upward. (c) Why didn’t the normal force enter into your solution?

Problem 7DQ Is it possible for a friction force to increase the mechanical energy of a system? If so, give examples.

Problem 7E BIO Human Energy vs. Insect Energy. For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.0-mm-long, 0.50-mg flea can reach a height of 20 cm in a single leap. (a) Ignoring air drag, what is the take-off speed of such a flea? (b) Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass. (c) If a 65-kg, 2.0-m-tall human could jump to the same height compared with his length as the flea jumps compared with its length, how high could the human jump, and what takeoff speed would the man need? (d) Most humans can jump no more than 60 cm from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a 65-kg person? (e) Where does the flea store the energy that allows it to make sudden leaps?

Problem 8DQ A woman bounces on a trampoline, going a little higher with each bounce. Explain how she increases the total mechanical energy.

Problem 8E An empty crate is given an initial push down a ramp, starting with speed ?v?0, and reaches the bottom with speed ?v? and kinetic energy ?K?. Some books are now placed in the crate, so that the total mass is quadrupled. The coefficient of kinetic fiction is constant and air resistance is negligible. Starting again with ?v?0 at the top of the ramp, what are the speed and kinetic energy at the bottom? Explain the reasoning behind your answers.

Problem 9DQ Fractured Physics. People often call their electric bill a power bill, yet the quantity on which the bill is based is expressed in kilowatt-hours. What are people really being billed for?

Problem 9E CP A small rock with mass 0.20 kg is released from rest at point A , which is at the top edge of a large, hemispherical bowl with radius R = 0.50 m (Fig. E7.9). Assume that the size of the rock is small compared to R, so that the rock can be treated as a particle, and assume that the rock slides rather than rolls. The work done by friction on the rock when it moves from point A to point B at the bottom of the bowl has magnitude 0.22 J. (a) Between points A and B, how much work is done on the rock by (i) the normal force and (ii) gravity? (b) What is the speed of the rock as it reaches point B? (c) Of the three forces acting on the rock as it slides down the bowl, which (if any) are constant and which are not? Explain. (d) Just as the rock reaches point B, what is the normal force on it due to the bottom of the bowl?

Problem 10DQ A rock of mass m? ? and a rock of mass 2?m? are both released from rest at the same height and feel no air resistance as they fall. Which statements about these rocks are true? (There may be more than one correct choice.) (a) Both have the same initial gravitational potential energy. (b) Both have the same kinetic energy when they reach the ground. (c) Both reach the ground with the same speed. (d) When it reaches the ground, the heavier rock has four times the kinetic energy of the lighter one. (e) When it reaches the ground, the heavier rock has four times the kinetic energy of the lighter one.

Problem 10E BIO Bone Fractures. The maximum energy that a bone can absorb without breaking depends on characteristics such as its cross-sectional area and elasticity. For healthy human leg bones of approximately 6.0 cm2 cross-sectional area, this energy has been experimentally measured to be about 200 J. (a) From approximately what maximum height could a 60-kg person jump and land rigidly upright on both feet without breaking his legs? (b) You are probably surprised at how small the answer to part (a) is. People obviously jump from much greater heights without breaking their legs. How can that be? What else absorbs the energy when they jump from greater heights? (?Hint:? How did the person in part (a) land? How do people normally land when they jump from greater heights?) (c) Why might older people be much more prone than younger ones to bone fractures from simple falls (such as a fall in the shower)?

Problem 11DQ On a friction-free ice pond, a hockey puck is pressed against (but not attached to) a fixed ideal spring, compressing the spring by a distance ?x?0. The maximum energy stored in the spring is ?U0 ? , the maximum speed the puck gains after being released is ?v?0, and its maximum kinetic energy is ?K?0. Now the puck is please so it compresses the spring twice as far as before. In this case, (a) what is the maximum potential energy stored in the spring (m terms of ?U?0), and (b) what are the puck’s maximum kinetic energy and speed (in terms of ?K?0 and ? ?0)?

Problem 11E You are testing a new amusement park roller coaster with an empty car of mass 120 kg. One part of the track is a vertical loop with radius 12.0 m. At the bottom of the loop (point A ) the car has speed 25.0 m/s, and at the top of the loop (point B) it has speed 8.0 m/s. As the car rolls from point A to point B, how much work is done by friction?

Problem 12DQ When people are cold, they often rub their hands together to warm up. How does doing this produce heat? Where does the heat come from?

Problem 12E Tarzan and Jane. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 m that makes an angle of 45o with the vertical, steps off his tree limb, and swings down and then up to Jane’s open arms. When he arrives, his vine makes an angle of 30o with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan’s speed just before he reaches Jane. Ignore air resistance and the mass of the vine.

Problem 13E A 10.0-kg microwave oven is pushed 8.00 m up the sloping surface of a loading ramp inclined at an angle of 36.9° above the horizontal, by a constant force with a magnitude 110 N and acting parallel to the ramp. The coefficient or kinetic friction between the oven and the ramp is 0.250. (a) What is the work done on the oven by the force ? (b) What is the work done on the oven by the friction force? (c) Compute the increase in potential energy for the oven. (d) Use your answers to parts (a), (b), and (c) co calculate the increase in the oven’s kinetic energy. (e) Use to calculate the acceleration of the oven. Assuming that the oven is initially at rest, use the acceleration to calculate the oven’s speed after traveling 8.00 m. From this, compute the increase in the oven’s kinetic energy, and compare it to the answer you got in part (d).

Problem 13DQ You often hear it said that most of our energy ultimately comes from the sun. Trace each of the following energies back to the sun: (a) the kinetic energy of a jet plane; (b) the potential energy gained by a mountain climber, (c) the electrical energy used to run a computer; (d) the electrical energy from a hydroelectric plant.

Problem 14DQ A box slides down a ramp and work is done on the box by the forces of gravity and friction. Can the work of each of these forces be expressed in terms of the change in a potential-energy function? For each force explain why or why not.

Problem 14E An ideal spring of negligible mass is 12.00 cm long when nothing is attached to it. When you hang a 3.15-kg weight from it, you measure its length to be 13.40 cm. If you wanted to store 10.0 J of potential energy in this spring, what would be its ?total? length? Assume that it continues to obey Hooke’s law.

Problem 15DQ In physical terms, explain why friction is a nonconservative force. Does it store energy for future use?

Problem 15E A force of 800 N stretches a certain spring a distance of 0.200 m. (a) What is the potential energy of the spring when it is stretched 0.200 m? (b) What is its potential energy when it is compressed 5.00 cm?

Problem 16DQ A compressed spring is clamped in its compressed position and then is dissolved in acid. What becomes of its potential energy?

Problem 16E BIO Tendons. Tendons are strong elastic fibers that attach muscles to bones. To a reasonable approximation, they obey Hooke’s law. In laboratory tests on a particular tendon, it was found that, when a 250-g object was hung from it, the tendon stretched 1.23 cm. (a) Find the force constant of this tendon in N/m. (b) Because of its thickness, the maximum tension this tendon can support without rupturing is 138 N. By how much can the tendon stretch without rupturing, and how much energy is stored in it at that point?

Problem 17DQ Since only changes in potential energy are important in any problem, a student decides to let the elastic potential energy of a spring be zero when the spring is stretched a distance x1. The student decides, therefore, to let Is this correct? Explain.

Problem 17E A spring stores potential energy U0 when it is compressed a distance x0 from its uncompressed length. (a) In terms of U0, how much energy does the spring store when it is compressed (i) twice as much and (ii) half as much? (b) In terms of x0, how much must the spring be compressed from its uncompressed length to store (i) twice as much energy and (ii) half as much energy?

Problem 18DQ Figure 7.22a shows the potential-energy function for the force Fx = -kx. Sketch the potential-energy function for the force Fx = +kx. For this force, is x = 0 a point of equilibrium? Is this equilibrium stable or unstable? Explain.

Problem 18E A slingshot will shoot a 10-g pebble 22.0 m straight up. (a) How much potential energy is stored in the slingshot’s rubber band? (b) With the same potential energy stored in the rubber band, how high can the slingshot shoot a 25-g pebble? (c) What physical effects did you ignore in solving this problem?

Figure 7.22b shows the potential-energy function associated with the gravitational force between an object and the earth. Use this graph to explain why objects always fall toward the earth when they are released.

Problem 19E A spring of negligible mass has force constant k = 1600 N/m. (a) How far must the spring be compressed for 3.20 J of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then drop a 1.20-kg book onto it from a height of 0.800 m above the top of the spring. Find the maximum distance the spring will be compressed.

A 1.20-kg piece of cheese is placed on a vertical spring of negligible mass and force constant k = 1800 N / m that is compressed 15.0 cm. When the spring is released, how high does the cheese rise from this initial position? (The cheese and the spring are not attached.)

Problem 20DQ For a system of two particles we often let the potential energy for the force between the particles approach zero as the separation of the particles approaches infinity. If this choice is made, explain why the potential energy at noninfinite separation is positive if the particles repel one another and negative if they attract.

Problem 21DQ Explain why the points x = A and x = - A in Fig. 7.23b are called ?turning points.? How are the values of E and U related at a turning point?

Problem 21E Consider the glider of Example 7.7 (Section 7.2) and Fig. 7.16. As in the example, the glider is released from rest with the spring stretched 0.100 m. What is the displacement x of the glider from its equilibrium position when its speed is 0.20 m/s? (You should get more than one answer. Explain why.)

Problem 23DQ The net force on a particle of mass m has the potential-energy function graphed in Fig. 7.24a. If the total energy is El, graph the speed v of the particle versus its position x. At what value of x is the speed greatest? Sketch v versus x if the total energy is

Problem 22DQ A particle is in ?neutral equilibrium? if the net force on it is zero and remains zero if the particle is displaced slightly in any direction. Sketch the potential-energy function near a point of neutral equilibrium for the case of one-dimensional motion. Give an example of an object in neutral equilibrium.

Problem 23E A 2.50-kg mass is pushed against a horizontal spring of force constant 25.0 N/cm on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 J of potential energy in it, the mass is suddenly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest acceleration of the mass, and when does it occur?

Problem 22E Consider the glider of Example 7.7 (Section 7.2) and Fig. 7.16. (a) As in the example, the glider is released from rest with the spring stretched 0.100 m. What is the speed of the glider when it returns to x = 0? (b) What must the initial displacement of the glider be if its maximum speed in the subsequent motion is to be 2.50 m/s?

Problem 24DQ The potential-energy function for a force where ? is a positive constant. What is the direction of

Problem 24E (a) For the elevator of Example 7.9 (Section 7.2), what is the speed of the elevator after it has moved downward 1.00 m from point 1 in Fig. 7.17? (b) When the elevator is 1.00 m below point 1 in Fig. 7.17, what is its acceleration?

Problem 25E You are asked to design a spring that will give a 1160-kg satellite a speed of 2.50 m/s relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of 5.00 g. The spring’s mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will all be negligible. (a) What must the force constant of the spring be? (b) What distance must the spring be compressed?

Problem 26E A 2.50-kg block on a horizontal floor is attached to a horizontal spring that is initially compressed 0.0300 m. The spring has force constant 840 N/m. The coefficient of kinetic friction between the floor and the block is µk = 0.40. The block and spring are released from rest, and the block slides along the floor. What is the speed of the block when it has moved a distance of 0.0200 m from its initial position? (At this point the spring is compressed 0.0100 m.)

Problem 27E A 10.0-kg box is pulled by a horizontal wire in a circle on a rough horizontal surface for which the coefficient of kinetic friction is 0.250. Calculate the work done by friction during one complete circular trip if the radius is (a) 2.00 m and (b) 4.00 m. (c) On the basis of the results you just obtained, would you say that friction is a conservative or nonconservative force? Explain.

Problem 28E A 75-kg roofer climbs a vertical 7.0-m ladder to the flat roof of a house. He then walks 12 m on the roof, climbs down another vertical 7.0-m ladder, and finally walks on the ground back to his starting point. How much work is done on him by gravity (a) as he climbs up; (b) as he climbs down; (c) as he walks on the roof and on the ground? (d) What is the total work done on him by gravity during this round trip? (e) On the basis of your answer to part (d), would you say that gravity is a conservative or nonconservative force? Explain.

Problem 29E A 0.60-kg book slides on a horizontal table. The kinetic friction force on the book has magnitude 1.8 N. (a) How much work is done on the book by friction during a displacement of 3.0 m to the left? (b) The book now slides 3.0 m to the right, returning to its starting point. During this second 3.0-m displacement, how much work is done on the book by friction? (c) What is the total work done on the book by friction during the complete round trip? (d) On the basis of your answer to part (c), would you say that the friction force is conservative or nonconservative? Explain.

Problem 30E CALC In an experiment, one of the forces exerted on a proton is where ? = 12 N/m2. (a) How much work does do when the proton moves along the straight-line path from the point (0.10 m, 0) to the point (0.10 m, 0.40 m)? (b) Along the straight-line path from the point (0.10 m, 0) to the point (0.30 m, 0)? (c) Along the straight-line path from the point (0.30 m, 0) to the point (0.10 m, 0)? (d) Is the force conservative? Explain. If is conservative, what is the potential-energy function for it? Let U = 0 when x = 0.

Problem 32E While a roofer is working on a roof that slants at 36° above the horizontal, he accidentally nudges his 85.0-N toolbox, causing it to start sliding downward from rest. If it starts 4.25 m from the lower edge of the roof, how fast will the toolbox be moving just as it reaches the edge of the roof if the kinetic friction force on it is 22.0 N?

Problem 33E A 62.0-kg skier is moving at 6.50 m/s on a frictionless, horizontal, snow-covered plateau when she encounters a rough patch 3.50 m long. The coefficient of kinetic friction between this patch and her skis is 0.300. After crossing the rough patch and returning to friction-free snow, she skis down an icy, frictionless hill 2.50 m high. (a) How fast is the skier moving when she gets to the bottom of the hill? (b) How much internal energy was generated in crossing the rough patch?

Problem 31E You and three friends stand at the corners of a square whose sides are 8.0 m long in the middle of the gym floor, as shown in Fig. You take your physics book and push it from one person to the other. The book has a mass of 1.5 kg, and the coefficient of kinetic friction between the book and the floor is ???k = 0.25. (a) The book slides from you to Beth and then from Beth to Carlos, along the lines connecting these people. What is the work done by friction during this displacement? (b) You slide the book from you to Carlos along the diagonal of the square. What is the work done by friction during this displacement? (c) You slide the book to Kim, who then slides it back to you. What is the total work done by friction during this motion of the book? (d) Is the friction force on the book conservative or nonconservative? Explain. Figure:

Problem 34E CALC The potential energy of a pair of hydrogen atoms separated by a large distance x is given by U(x) = -C6/x6, where C6 is a positive constant. What is the force that one atom exerts on the other? Is this force attractive or repulsive?

Problem 35E A force parallel to the ?x?-axis acts on a particle moving along the ?x?-axis. This force produces potential energy ?U?(?x?) given by ?U?(?x?) = ??x?4, where ??? = 1.20 J/m4. What is the force (magnitude and direction) when the particles is at ?x? = ?0.0800 m?

Problem 36E CALC An object moving in the xy-plane is acted on by a conservative force described by the potential-energy function U(x , y) = ?[(1/x2) + (1/y2)], where ? is a positive constant. Derive an expression for the force expressed in terms of the unit vectors

Problem 38E A marble moves along the x -axis. The potential-energy function is shown in Fig. E7.36. (a) At which of the labeled x-coordinates is the force on the marble zero? (b) Which of the labeled x-coordinates is a position of stable equilibrium? (c) Which of the labeled x-coordinates is a position of unstable equilibrium?

Problem 37E CALC A small block with mass 0.0400 kg is moving in the xy -plane. The net force on the block is described by the potential- energy function U(x , y) = (5.80 J/m2)x2 – (3.60 J/m3)y3. What are the magnitude and direction of the acceleration of the block when it is at the point (x = 0.300 m, y = 0.600 m)?

CALC The potential energy of two atoms in a diatomic molecule is approximated by \(U(r)=a / r^{12}-b / r^{6}\) where r is the spacing between atoms and a and b are positive constants. (a) Find the force F (r) on one atom as a function of r. Draw two graphs: one of versus U (r) versus r and one of F (r) versus r. (b) Find the equilibrium distance between the two atoms. Is this equilibrium stable? (c) Suppose the distance between the two atoms is equal to the equilibrium distance found in part (b). What minimum energy must be added to the molecule to dissociate it—that is, to separate the two atoms to an infinite distance apart? This is called the dissociation energy of the molecule. (d) For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is \(1.13 \times 10^{-10} \mathrm{m}\) and the dissociation energy is \(1.54 \times 10^{-18} \mathrm{J}\) per molecule. Find the values of the constants a and b.

Problem 40P Two blocks with different masses are attached to either end of a light rope that passes over a light, frictionless pulley suspended from the ceiling. The masses are released from rest and the more massive one starts to descend. After this block has descended 1.20 m, its speed is 3.00 m/s. If the total mass of the two blocks is 15.0 kg, what is the mass of each block?

Problem 41P At a construction site, a 65.0-kg bucket of concrete hangs from a light (but strong) cable that passes over a light, friction-free pulley and is connected to an 80.0-kg box on a horizontal roof (Fig. P7.37). The cable pulls horizontally on the box, and a 50.0-kg bag of gravel rests on top of the box. The coefficients of friction between the box and roof are shown. (a) Find the friction force on the bag of gravel and on the box. (b) Suddenly a worker picks up the bag of gravel. Use energy conservation to find the speed of the bucket after it has descended 2.00 m from rest. (Use Newton’s laws to check your answer.)

Problem 42P A 2.00-kg block is pushed against a spring with negligible mass and force constant k = 400 N/m, compressing it 0.220 m. When the block is released, it moves along a frictionless, horizontal surface and then up a frictionless incline with slope 37.0o (Fig. P7.40). (a) What is the speed of the block as it slides along the horizontal surface after having left the spring? (b) How far does the block travel up the incline before starting to slide back down?

Problem 43P A block with mass 0.50 kg is forced against a horizontal spring of negligible mass, compressing the spring a distance of 0.20 m (Fig. P7.39). When released, the block moves on a horizontal tabletop for 1.00 m before coming to rest. The force constant k is 100 N/m. What is the coefficient of kinetic friction µk between the block and the tabletop?

Problem 44P On a horizontal surface, a crate with mass 50.0 kg is placed against a spring that stores 360 J of energy. The spring is released, and the crate slides 5.60 m before coming to rest. What is the speed of the crate when it is 2.00 m from its initial position?

Problem 45P A 350-kg roller coaster car starts from rest at point A and slides down a frictionless loop-the-loop (Fig. P7.41). (a) How fast is this roller coaster car moving at point B? (b) How hard does it press against the track at point B?

Problem 46P CP Riding a Loop-the-Loop. A car in an amusement park ride rolls without friction around a track (Fig. P7.42). The car starts from rest at point A at a height h above the bottom of the loop. Treat the car as a particle. (a) What is the minimum value of h (in terms of R) such that the car moves around the loop without falling off at the top (point B)? (b) If ?h? = 3.50 R and ?R? = 14.0 m, compute the speed, radial acceleration, and tangential acceleration of the passengers when the car is at point C , which is at the end of a horizontal diameter. Show these acceleration components in a diagram, approximately to scale.

Problem 47P A 2.0-kg piece of wood slides on a curved surface (Fig. P7.43). The sides of the surface are perfectly smooth, but the rough horizontal bottom is 30 m long and has a kinetic friction coefficient of 0.20 with the wood. The piece of wood starts from rest 4.0 m above the rough bottom. (a) Where will this wood eventually come to rest? (b) For the motion from the initial release until the piece of wood comes to rest, what is the total amount of work done by friction?

Problem 48P Up and Down the Hill. A 28-kg rock approaches the foot of a hill with a speed of 15 m/s. This hill slopes upward at a constant angle of 40.0 ? above the horizontal. The coefficients of static and kinetic friction between the hill and the rock are 0.75 and 0.20, respectively. (a) Use energy conservation to find the maximum height above the foot of the hill reached by the rock. (b) Will the rock remain at rest at its highest point, or will it slide back down the hill? (c) If the rock does slide back down, find its speed when it returns to the bottom of the hill.

Problem 49P A 15.0-kg stone slides down a snow-covered hill (Fig. P7.45), leaving point A at a speed of 10.0 m/s. There is no friction on the hill between points A and B, but there is friction on the level ground at the bottom of the hill, between B and the wall. After entering the rough horizontal region, the stone travels 100 m and then runs into a very long, light spring with force constant 2.00 N/m. The coefficients of kinetic and static friction between the stone and the horizontal ground are 0.20 and 0.80, respectively. (a) What is the speed of the stone when it reaches point B? (b) How far will the stone compress the spring? (c) Will the stone move again after it has been stopped by the spring?

Problem 50P CP A 2.8-kg block slides over the smooth, icy hill shown in Fig. P7.46. The top of the hill is horizontal and 70 m higher than its base. What minimum speed must the block have at the base of the 70-m hill to pass over the pit at the far (right-hand) side of that hill?

Problem 51P Bungee Jump. A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude ?kx?. Your father-in-law (mass 95.0 kg) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 380.0 N. When you do this, what distance will the bungee cord that you should select have stretched?

Problem 52P Ski Jump Ramp. You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height ?h? from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of 2.0 m/s as they reach the gate. For safety, the skiers should have a speed no higher than 30.0 m/s when they reach the bottom of the ramp. You determine that for a 85.0-kg skier with good form, friction and air resistance will do total work of magnitude 4000 J on him during his run down the ramp. What is the maximum height ?h? for which the maximum safe speed will not be exceeded?

Problem 54P You are designing a delivery ramp for crates containing exercise equipment. The 1470-N crates will move at 1.8 m/s at the top of a ramp that slopes downward at 22.0o. The ramp exerts a 515-N kinetic friction force on each crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of 5.0 m along the ramp. Once stopped, a crate must not rebound back up the ramp. Calculate the largest force constant of the spring that will be needed to meet the design criteria.

Problem 53P The Great Sandini is a 60-kg circus performer who is shot from a cannon (actually a spring gun). You don’t find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100 N/m that he will compress with a force of 4400 N. The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40 N during the 4.0 m he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.5 m above his initial rest position?

Problem 55P A system of two paint buckets connected by a light-weight rope is released from rest with the 12.0-kg bucket 2.00 m above the floor (Fig. P7.51). Use the principle of conservation of energy to find the speed with which this bucket strikes the floor. Ignore friction and the mass of the pulley.

A 1500-kg rocket is to be launched with an initial upward speed of 50.0 m/s. In order to assist its engines, the engineers will start it from rest on a ramp that rises 53° above the horizontal (Fig. P7.50). At the bottom, the ramp turns upward and launches the rocket vertically. The engines provide a constant forward thrust of 2000 N, and friction with the ramp surface is a constant 500 N. How far from the base of the ramp should the rocket start, as measured along the surface of the ramp?

Legal Physics. In an auto accident, a car hit a pedestrian and the driver then slammed on the brakes to stop the car. During the subsequent trial, the driver’s lawyer claimed that he was obeying the posted speed limit, but that the legal speed was too high to allow him to see and react to the pedestrian in time. You have been called in as the state’s expert witness. Your investigation of the accident found that the skid marks made while the brakes were applied were 280 ft long, and the tread on the tires produced a coefficient of kinetic friction of 0.30 with the road. (a) In your testimony in court, will you say that the driver was obeying the posted speed? You must be able to back up your conclusion with clear reasoning because one of the lawyers will surely cross-examine you. (b) If the driver’s speeding ticket were $10 for each mile per hour he was driving above the posted speed limit, would he have to pay a fine? If so, how much would it be?

Problem 58P A wooden rod of negligible mass and length 80.0 cm is pivoted about a horizontal axis through its center. A white rat with mass 0.500 kg clings to one end of the stick, and a mouse with mass 0.200 kg clings to the other end. The system is released from rest with the rod horizontal. If the animals can manage to hold on, what are their speeds as the rod swings through a vertical position?

Problem 59P CP A 0.300-kg potato is tied to a string with length 2.50 m, and the other end of the string is tied to a rigid support. The potato is held straight out horizontally from the point of support, with the string pulled taut, and is then released. (a) What is the speed of the potato at the lowest point of its motion? (b) What is the tension in the string at this point?

Problem 60P These results are from a computer simulation for a batted baseball with mass 0.145 kg, including air resistance: How much work did the air do on the baseball (a) as the ball moved from its initial position to its maximum height, and (b) as the ball moved from its maximum height back to the starting elevation? (c) Explain why the magnitude of the answer in part (b) is smaller than the magnitude of the answer in part (a).

Problem 61P Down the Pole. A fireman of mass m ? ? slides a distanc? ? down a pole. He starts from rest. He moves as fast at the bottom as if he had stepped off a platform a distance ?h? ? ?d? above the ground and descended with negligible air resistance. (a) What average friction force did the fireman exert on the pole? Does your answer make sense in the special case of ?h? = ?d? and ?h? = 0? (b) Find a numerical value for the average fiction force 75 kg fireman exerts, for ?d? = 2.5 m and h? ? = 1.0 m. (c) In terms of ?g?, ?h?? ?. what is the speed of the fireman when he is a distance ?y? above the bottom of the pole?

Problem 62P A 60.0-kg skier starts from rest at the top of a ski slope 65.0 m high. (a) If friction forces do - 10.5 kJ of work on her as she descends, how fast is she going at the bottom of the slope? (b) Now moving horizontally, the skier crosses a patch of soft snow where µk = 0.20. If the patch is 82.0 m wide and the average force of air resistance on the skier is 160 N, how fast is she going after crossing the patch? (c) The skier hits a snowdrift and penetrates 2.5 m into it before coming to a stop. What is the average force exerted on her by the snowdrift as it stops her?

CP A skier starts at the top of a very large, frictionless snowball, with a very small initial speed, and skis straight down the side (Fig. P7.55). At what point does she lose contact with the snowball and fly off at a tangent? That is, at the instant she loses contact with the snowball, what angle \(\alpha\) does a radial line from the center of the snowball to the skier make with the vertical?

Problem 64P A ball is thrown upward with an initial velocity of 15 m/s at an angle of 60.0° above the horizontal. Use energy conservation to find the ball’s greatest height above the ground.

Problem 65P In a truck-loading station at a post office, a small 0.200-kg package is released from rest at point A on a track that is one-quarter of a circle with radius 1.60 m (Fig. P7.57). The size of the package is much less than 1.60 m, so the package can be treated as a particle. It slides down the track and reaches point B with a speed of 4.80 m/s. From point B, it slides on a level surface a distance of 3.00 m to point C, where it comes to rest. (a) What is the coefficient of kinetic friction on the horizontal surface? (b) How much work is done on the package by friction as it slides down the circular arc from A to B?

Problem 66P A truck with mass m has a brake failure while going down an icy mountain road of constant downward slope angle ? (Fig. P7.58). Initially the truck is moving downhill at speed v0. After careening downhill a distance L with negligible friction, the truck driver steers the runaway vehicle onto a runaway truck ramp of constant upward slope angle ?. The truck ramp has a soft sand surface for which the coefficient of rolling friction is µr. What is the distance that the truck moves up the ramp before coming to a halt? Solve by energy methods.

Problem 68P CP A sled with rider having a combined mass of 125 kg travels over a perfectly smooth icy hill (Fig. P7.60). How far does the sled land from the foot of the cliff?

Problem 67P CALC A certain spring found not to obey Hooke’s law exerts a restoring force Fx(x) = -?x - ?x2 if it is stretched or compressed, where ? = 60.0 N/m and ? = 18.0 Nm2. The mass of the spring is negligible. (a) Calculate the potential-energy function U(x) for this spring. Let U = 0 when x = 0. (b) An object with mass 0.900 kg on a frictionless, horizontal surface is attached to this spring, pulled a distance 1.00 m to the right (the +x-direction) to stretch the spring, and released. What is the speed of the object when it is 0.50 m to the right of the x = 0 equilibrium position?

Problem 69P A 0.150-kg block of ice is placed against a horizontal, compressed spring mounted on a horizontal tabletop that is 1.20 m above the floor. The spring has force constant 1900 N/m and is initially compressed 0.045 m. The mass of the spring is negligible. The spring is released, and the block slides along the table, goes off the edge, and travels to the floor. If there is negligible friction between the block of ice and the tabletop, what is the speed of the block of ice when it reaches the floor?

Problem 70P A 3.00-kg block is connected to two ideal horizontal springs having force constants k1 = 25.0 N/cm and k2 = 20.0 N/cm (Fig. P7.62). The system is initially in equilibrium on a horizontal, frictionless surface. The block is now pushed 15.0 cm to the right and released from rest. (a) What is the maximum speed of the block? Where in the motion does the maximum speed occur? (b) What is the maxi-mum compression of spring 1?

Problem 71P An experimental apparatus with mass ?m? is placed on a vertical spring of negligible mass and pushed down until the spring is compressed a distance ?x. The apparatus is then released and reaches its maximum height at a distance l?1 above the point where it is released. The apparatus is not attached to the spring, and at its maximum height it is no longer in contact with the spring. The maximum magnitude or acceleration the apparatus can have without being damaged is a, where ?a? > ?g?. (a) What should the force constant of the spring be? (b) What distance ?x? must the spring be compressed initially?

Problem 72P If a fish is attached to a vertical spring and slowly lowered to its equilibrium position, it is found to stretch the spring by an amount ?d?. If the same fish is attached to the end of the un-stretched spring and then allowed to fall from rest, through what maximum distance does it stretch the spring? (?Hint: Calculate the force constant of the spring in terms of the distance ?d? and the mass m? ? of the fish.)

Problem 73P CALC A 3.00-kg fish is attached to the lower end of a vertical spring that has negligible mass and force constant 900 N/m. The spring initially is neither stretched nor compressed. The fish is released from rest. (a) What is its speed after it has descended 0.0500 m from its initial position? (b) What is the maximum speed of the fish as it descends?

Problem 74P A basket of negligible weight hangs from a vertical spring scale of force constant 1500 N/m. (a) If you suddenly put a 3.0-kg adobe brick in the basket, find the maximum distance that the spring will stretch. (b) If, instead, you release the brick from 1.0 m above the basket, by how much will the spring stretch at its maximum elongation?

Problem 75P A 0.500-kg block, attached to a spring with length 0.60 m and force constant 40.0 N/m, is at rest with the back of the block at point A on a frictionless, horizontal air table (Fig. P7.69). The mass of the spring is negligible. You move the block to the right along the surface by pulling with a constant 20.0-N horizontal force. (a) What is the block’s speed when the back of the block reaches point B, which is 0.25 m to the right of point A? (b) When the back of the block reaches point B, you let go of the block. In the subsequent motion, how close does the block get to the wall where the left end of the spring is attached?

Problem 76P Fraternity Physics. The brothers of Iota Eta Pi fraternity build a platform, supported at all four corners by vertical springs, in the basement of their frat house. A brave fraternity brother wearing a football helmet stands in the middle of the platform; his weight compresses the springs by 0.18 m. Then four of his fraternity brothers, pushing down at the corners of the platform, compress the springs another 0.53 m until the top of the brave brother’s helmet is 0.90 m below the basement ceiling. They then simultaneously release the platform. You can ignore the masses of the springs and platform. (a) When the dust clears, the fraternity asks you to calculate their fraternity brother’s speed just before his helmet hit the flimsy ceiling. (b) Without the ceiling how high would he have gone? (c) In discussing their probation the dean of students suggests that the next time they try this, they do it outdoors on another planet. Would the answer to part (b) be the same if this stunt were performed on a planet with a different value of ?g?? Assume that the fraternity brothers push the platform down 0.53 m as before. Explain your reasoning.

Problem 77P CP A small block with mass 0.0500 kg slides in a vertical circle of radius R = 0.800 m on the inside of a circular track. There is no friction between the track and the block. At the bottom of the block’s path, the normal force the track exerts on the block has magnitude 3.40 N. What is the magnitude of the normal force that the track exerts on the block when it is at the top of its path?

Problem 78P CP A small block with mass 0.0400 kg slides in a vertical circle of radius R = 0.500 m on the inside of a circular track. During one of the revolutions of the block, when the block is at the bottom of its path, point A, the normal force exerted on the block by the track has magnitude 3.95 N. In this same revolution, when the block reaches the top of its path, point B, the normal force exerted on the block has magnitude 0.680 N. How much work is done on the block by friction during the motion of the block from point A to point B?

Problem 79P A hydroelectric dam holds back a lake of surface area 3.0 × 106 m2 that has vertical sides below the water level. The water level in the lake is 150 m above the base of the dam. When the water passes through turbines at the base of the dam, its mechanical energy is converted to electrical energy with 90% efficiency. (a) lf gravitational potential energy is taken to be zero at the base of the dam, how much energy is stored in the top meter of the water in the lake? The density of water is 1000 kg/m3. (b) What volume of water must pass through the dam to produce 1000 kilowatt-hours of electrical energy? What distance does the level of water in the lake fall when this much water passes through the dam?

Problem 80P How much total energy is stored in the lake in Problem? As in that problem, take the gravitational potential energy to be zero at the base of the dam. Express your answer in joules and in kilowatt-hours. (?Hint?: Break the lake up into infinitesimal horizontal layers of thickness ?dy?, and integrate to find the total potential energy.) Problem: A hydroelectric dam holds back a lake of surface area 3.0 × 106 m2 that has vertical sides below the water level. The water level in the lake is 150 m above the base of the dam. When the water passes through turbines at the base of the dam, its mechanical energy is converted to electrical energy with 90% efficiency. (a) lf gravitational potential energy is taken to be zero at the base of the dam, how much energy is stored in the top meter of the water in the lake? The density of water is 1000 kg/m3. (b) What volume of water must pass through the dam to produce 1000 kilowatt-hours of electrical energy? What distance does the level of water in the lake fall when this much water passes through the dam?

Problem 81P A wooden block with mass 1.50 kg is placed against a compressed spring at the bottom of an incline of slope 30.0o (point A). When the spring is released, it projects the block up the incline. At point B, a distance of 6.00 m up the incline from A , the block is moving up the incline at 7.00 m/s and is no longer in contact with the spring. The coefficient of kinetic friction between the block and the incline is µk = 0.50. The mass of the spring is negligible. Calculate the amount of potential energy that was initially stored in the spring.

Problem 82P CP Pendulum. A small rock with mass 0.12 kg is fastened to a massless string with length 0.80 m to form a pendulum. The pendulum is swinging so as to make a maximum angle of 45o with the vertical. Air resistance is negligible. (a) What is the speed of the rock when the string passes through the vertical position? What is the tension in the string (b) when it makes an angle of 45o with the vertical, (c) as it passes through the vertical?

Problem 83P CALC A cutting tool under microprocessor control has several forces acting on it. One force is a force in the negative y -direction whose magnitude depends on the position of the tool. For ? = 2.50 N/m, consider the displacement of the tool from the origin to the point (x = 3.00 m, y = 3.00m). (a) Calculate the work done on the tool by if this displacement is along the straight line y = x that connects these two points. (b) Calculate the work done on the tool by if the tool is first moved out along the x-axis to the point (x = 3.00 m, y = 0) and then moved parallel to the y -axis to the point (x = 3.00 m, y = 3.00 m). (c) Compare the work done by F S along these two paths. Is conservative or nonconservative? Explain.

Problem 84P (a) Is the force where C? ? is a negative constant with units of N/m2, conservative or nonconservative? Justify your answer. (b) Is the force where ?C? is a negative constant with units of N/m2, conservative or nonconservative? Justify your answer.

Problem 85P An object has several forces acting on it. One force is a force in the x?-direction whose magnitude depends on the position of the object. (See problem 6.98). The constant is ??? = 2.00 N/m2. The object moves along the following path; (1) It starts at the origin and moves along the ?y?-axis to the point ?x? = 0, ?y? = 1.50 m; (2) it moves parallel to the ?x?-axis to the point ?x? = 1.50 m, ?y? = 1.50 m; (3) it moves parallel to the ?y?-axis to the point ?x? = 1.50, ?y? = 0; (4) it moves parallel to the ?x?-axis back to the origin. (a) Sketch this path in the ?xy-?plane. (b) Calculate the work done on the object by for each leg of the path and for the complete round trip. (c) Is conservative or nonconservative? Explain.

Problem 86P A particle moves along the x-axis while acted on by a single conservative force parallel to the x-axis. The force corresponds to the potential-energy function graphed in ?Fig. P7.76?. The particle is released from rest at point A. (a) What is the direction of the force on the particle when it is at point A? (b) At point B? (c) At what value of x is the kinetic energy of the particle a maximum? (d) What is the force on the particle when it is at point C? (e) What is the largest value of x reached by the particle during its motion? (f) What value or values of x correspond to points of stable equilibrium? (g) Of unstable equilibrium?