Two masses are connected by a string as shown inFig. 8-34.

A spring has a spring constant k of 82.0 N/m. How much must this spring be compressed to store 35.0 J of potential energy?

A 6.0-kg monkey swings from one branch to another 1.3 m higher. What is the change in gravitational potential energy?

A spring with k = 63 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?

A 56.5-kg hiker starts at an elevation of 1270 m and climbs to the top of a 2660-m peak, (a) What is the hikers change in potential energy? (b) What is the minimum work required of the hiker? (c) Can the actual work done be greater than this? Explain.

A 1.60-m tall person lifts a 1.95-kg book off the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to (a) the ground, and (b) the top of the persons head? (c) How is the work done by the person related to the answers in parts (a) and (6)?

A 1200-kg car rolling on a horizontal surface has speed v = 75 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring stiffness constant of the spring?

A particular spring obeys the force law F = (kx + ax3 + bxA) i. (a) Is this force conservative? Explain why or why not. (b) If it is conservative, determine the form of the potential energy function.

If U = 3x2 + 2xy + 4y2z, what is the force, F?

A particle is constrained to move in one dimension along the x axis and is acted upon by a force given by F(*) = XJ where k is a constant with units appropriate to the SI system. Find the potential energy function U(x), if U is arbitrarily defined to be zero at x = 2.0 m, so that U(2.0 m) = 0.

A particle constrained to move in one dimension is subject to a force F(x) that varies with position x as F(*) = A sm (k x )i where A and k are constants. What is the potential energy function U(x), if we take U = 0 at the point x = 0?

A novice skier, starting from rest, slides down a frictionless 13.0 incline whose vertical height is 125 m. How fast is she going when she reaches the bottom?

Jane, looking for Tarzan, is running at top speed (5.0 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?

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

A sled is initially given a shove up a frictionless 23.0 incline. It reaches a maximum vertical height 1.12 m higher than where it started. What was its initial speed?

A 55-kg bungee jumper leaps from a bridge. She is tied to a bungee cord that is 12 m long when unstretched, and falls a total of 31 m. (a) Calculate the spring constant k of the bungee cord assuming Hookes law applies. (b) Calculate the maximum acceleration she experiences.

A 72-kg trampoline artist jumps vertically upward from the top of a platform with a speed of 4.5 m/s. (a) How fast is he going as he lands on the trampoline, 2.0 m below (Fig. 8-31)? (b) If the trampoline behaves like a spring of spring constant 5.8 X 104N/m, how far does he depress it? FIGURE 8-31 Problem 16.

The total energy E of an object of mass m that moves in one dimension under the influence of only conservative forces can be written as E = \m v 2 + U. 2 Use conservation of energy, dE/dt = 0, to predict Newtons second law.

A 0.40-kg ball is thrown with a speed of 8.5 m/s at an upward angle of 36. {a) What is its speed at its highest point, and (b) how high does it go? (Use conservation of energy.)

A vertical spring (ignore its mass), whose spring constant is 875 N/m, is attached to a table and is compressed down by 0.160 m. (a) What upward speed can it give to a 0.380-kg ball when released? (b) How high above its original position (spring compressed) will the ball fly?

A roller-coaster car shown in Fig. 8-32 is pulled up to point 1 where it is released from rest. Assuming no friction, calculate the speed at points 2, 3, and 4.FIGURE 8-32 Problems 20 and 34.

When a mass m sits at rest on a spring, the spring is compressed by a distance d from its undeformed length (Fig. 8-33a). Suppose instead that the mass is released from rest when it barely touches the undeformed spring (Fig. 8-33b). Find the distance D that the spring is compressed before it is able to stop the mass. Does D = d l If not, why not?FIGURE 8-33 Problem 21.

Two masses are connected by a string as shown in Fig. 8-34. Mass mA = 4.0 kg rests on a frictionless inclined plane, while mB = 5.0 kg is initially held at a height of h = 0.75 m above the floor, (a) If mB is allowed to fall, what will be the resulting acceleration of the masses? (b) If the masses were initially at rest, use the kinematic equations (Eqs. 2-12) to find their velocity just before raB hits the floor, (c) Use conservation of energy to find the velocity of the masses just before raB hits the floor. You should get the same answer as in part (b).FIGURE 8-34 Problem 22.

A block of mass m is attached to the end of a spring (spring stiffness constant k ), Fig. 8-35. The mass is given an initial displacement x0 from equilibrium, and an initial speed v0. Ignoring friction and the mass of the spring, use energy methods to find (a) its maximum speed, and (b) its maximum stretch from equilibrium, in terms of the given quantities.FIGURE 8-35 Problems 23,37, and 38.

A cyclist intends to cycle up a 9.50 hill whose vertical height is 125 m. The pedals turn in a circle of diameter 36.0 cm. Assuming the mass of bicycle plus person is 75.0 kg, (a) calculate how much work must be done against gravity. (b) If each complete revolution of the pedals moves the bike 5.10 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.

A pendulum 2.00 m long is released (from rest) at an angle 0O = 30.0 (Fig. 8-14). Determine the speed of the 70.0-g bob: (a) at the lowest point (0 = 0); (b) at 0 = 15.0, (c) at 0 = -15.0 (i.e., on the opposite side). (d) Determine the tension in the cord at each of these three points. (e) If the bob is given an initial speed v0 = 1.20 m/s when released at 6 = 30.0, recalculate the speeds for parts (a), (b), and (c).

What should be the spring constant A: of a spring designed to bring a 1200-kg car to rest from a speed of 95 km/h so that the occupants undergo a maximum acceleration of 5.0 g?

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

A skier of mass m starts from rest at the top of a solid sphere of radius r and slides down its frictionless surface. (a) At what angle 0 (Fig. 8-36) will the skier leave the sphere? (b) If friction were present, would the skier fly off at a greater or lesser angle? FIGURE 8-36 Problem 28.

Two railroad cars, each of mass 56,000 kg, are traveling 95 km/h toward each other. They collide head-on and come to rest. How much thermal energy is produced in this collision?

A 16.0-kg child descends a slide 2.20 m high and reaches the bottom with a speed of 1.25 m/s. How much thermal energy due to friction was generated in this process?

A ski starts from rest and slides down a 28 incline 85 m long, (a) If the coefficient of friction is 0.090, what is the skis 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.

A 145-g baseball is dropped from a tree 14.0 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?

A 96-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 0.25. What is the final speed of the crate?

Suppose the roller-coaster car in Fig. 8-32 passes point 1 with a speed of 1.70 m/s. If the average force of friction is equal to 0.23 of its weight, with what speed will it reach point 2? The distance traveled is 45.0 m.

A skier traveling 9.0 m/s reaches the foot of a steady upward 19 incline and glides 12 m up along this slope before coming to rest. What was the average coefficient of friction?

Consider the track shown in Fig. 8-37. The section AB is one quadrant of a circle of radius 2.0 m and is frictionless. B to C is a horizontal span 3.0 m long with a coefficient of kinetic friction = 0.25. The section CD under the spring is frictionless. A block of mass 1.0 kg is released from rest at A. After sliding on the track, it compresses the spring by 0.20 m. Determine: (a) the velocity of the block at point B; (b) the thermal energy produced as the block slides from B to C; (c) the velocity of the block at point C; (d) the stiffness constant k for the spring.FIGURE 8-37 Problem 36.

A 0.620-kg wood block is firmly attached to a very light horizontal spring (k = 180 N/m) as shown in Fig. 8-35. This block-spring system, when compressed 5.0 cm and released, stretches out 2.3 cm beyond the equilibrium position before stopping and turning back. What is the coefficient of kinetic friction between the block and the table?

A 180-g wood block is firmly attached to a very light horizontal spring, Fig. 8-35. The block can slide along a table where the coefficient of friction is 0.30. A force of 25 N compresses the spring 18 cm. If the spring is released from this position, how far beyond its equilibrium position will it stretch on its first cycle?

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 balls speed just before and just after the bounce? (c) Where did the energy go?

A 56-kg skier starts from rest at the top of a 1200-mlong trail which drops a total of 230 m from top to bottom. At the bottom, the skier is moving 11.0 m/s. How much energy was dissipated by friction?

How much does your gravitational energy change when you jump as high as you can (say, 1.0 m)?

A spring (k = 75 N/m) has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.50 m and a mass of 2.0 kg is placed at its free end on a frictionless slope which makes an angle of 41 with respect to the horizontal (Fig. 8-38). The spring is then released, (a) If the mass is not attached to the spring, how far up the slope will the mass move before coming to rest? (b) If the mass is attached to the spring, how far up the slope will the mass move before coming to rest? (c) Now the incline has a coefficient of kinetic friction /%. If the block, attached to the spring, is observed to stop just as it reaches the springs equilibrium position, what is the coefficient of friction /%?FIGURE 8-38 Problem 42.

2.0-kg block slides along a horizontal surface with a coefficient of kinetic friction fjLk = 0.30. The block has a speed v = 1.3 m/s when it strikes a massless spring head-on (as in Fig. 8-18). (a) If the spring has force constant k = 120 N/m, how far is the spring compressed? (b) What minimum value of the coefficient of static friction, /jls , will assure that the spring remains compressed at the maximum compressed position? (c) If /jls is less than this, what is the speed of the block when it detaches from the decompressing spring? [Hint. Detachment occurs when the spring reaches its natural length (x = 0); explain why.]

Early test flights for the space shuttle used a glider (mass of 980 kg including pilot). After a horizontal launch at 480 km/h at a height of 3500 m, the glider eventually landed at a speed of 210 km/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 angle of 12 to the Earths surface?

For a satellite of mass ras in a circular orbit of radius rs around the Earth, determine (a) its kinetic energy K, (b) its potential energy U (U = 0 at infinity), and (c) the ratio K/U.

Jill and her friends have built a small rocket that soon after lift-off reaches a speed of 850 m/s. How high above the Earth can it rise? Ignore air friction.

The escape velocity from planet A is double that for planet B. The two planets have the same mass. What is the ratio of their radii, rA/rB?

Show that Eq. 8-16 for gravitational potential energy reduces to Eq. 8-2, A U = mg(y2 yi), for objects near the surface of the Earth.

Determine the escape velocity from the Sun for an object (a) at the Suns surface (r = 7.0 X 105km, M = 2.0 X IO30 kg), and (b) at the average distance of the Earth (1.50 X 108 km). Compare to the speed of the Earth in its orbit.

Two Earth satellites, A and B, each of mass m = 950 kg, are launched into circular orbits around the Earths center. Satellite A orbits at an altitude of 4200 km, and satellite B orbits at an altitude of 12,600 km. (a) What are the potential energies of the two satellites? (b) What are the kinetic energies of the two satellites? (c) How much work would it require to change the orbit of satellite A to match that of satellite B?

Show that the escape velocity for any satellite in a circular orbit is V2 times its velocity.

(a) Show that the total mechanical energy of a satellite (mass m) orbiting at a distance r from the center of the Earth (mass ME) is 1 GmME E = ~ 2 ~r ' if U = 0 at r = oo. (b) Show that although friction causes the value of E to decrease slowly, kinetic energy must actually increase if the orbit remains a circle.

Take into account the Earths rotational speed (1 rev/day) and determine the necessary speed, with respect to Earth, for a rocket to escape if fired from the Earth at the equator in a direction (a) eastward; (b) westward; (c) vertically upward.

(a) Determine a formula for the maximum height h that a rocket will reach if launched vertically from the Earths surface with speed vQ (< vesc). Express in terms of v0,rE, ME, and G. (b) How high does a rocket go if v0 = 8.35 km/s? Ignore air resistance and the Earths rotation.

(a) Determine the rate at which the escape velocity from the Earth changes with distance from the center of the Earth, dvesc/dr. (b) Use the approximation Av ~ (dv/dr) Ar to determine the escape velocity for a spacecraft orbiting the Earth at a height of 320 km.

A meteorite has a speed of 90.0 m/s when 850 km above the Earth. It is falling vertically (ignore air resistance) and strikes a bed of sand in which it is brought to rest in 3.25 m. (a) What is its speed just before striking the sand? (b) How much work does the sand do to stop the meteorite (mass = 575 kg)? (c) What is the average force exerted by the sand on the meteorite? (d) How much thermal energy is produced?

How much work would be required to move a satellite of mass m from a circular orbit of radius r\ = 2rE about the Earth to another circular orbit of radius r2 = 3rEl (rE is the radius of the Earth.)

(a) Suppose we have three masses, m i,m 2, and m3, that initially are infinitely far apart from each other. Show that the work needed to bring them to the positions shown in Fig. 8-39 is W G ( ml m2 + minis + m2m3\ V r12 r13 r23 (b) Can we say that this formula also gives the potential energy of the system, or the potential energy of one or two of the objects? (c) Is W equal to the binding energy of the systemthat is, equal to the energy required to separate the components by an infinite distance? Explain. FIGURE 8-39 Problem 58.

A NASA satellite has just observed an asteroid that is on a collision course with the Earth. The asteroid has an estimated mass, based on its size, of 5 X 109kg. It is approaching the Earth on a head-on course with a velocity of 660 m/s relative to the Earth and is now 5.0 X 106km away. With what speed will it hit the Earths surface, neglecting friction with the atmosphere?

A sphere of radius has a concentric spherical cavity of radius r2 (Fig. 8-40). Assume this spherical shell of thickness ri - r2 is uniform and has a total mass M. Show that the gravitational potential energy of a mass m at a distance r from the center of the shell (r > Tx) is given by U = - GmM FIGURE 8-40 Problem 60.

To escape the solar system, an interstellar spacecraft must overcome the gravitational attraction of both the Earth and Sun. Ignore the effects of other bodies in the solar system. (a) Show that the escape velocity is v = \ / v E + (vs - Vq)2 = 16.7 km/s, where: vE is the escape velocity from the Earth (Eq. 8-19); vs = V2GMs/r8E is the escape velocity from the gravitational field of the Sun at the orbit of the Earth but far from the Earths influence (rSE is the Sun-Earth distance); and v0 is the Earths orbital velocity about the Sun. (b) Show that the energy required is 1.40 X 108J per kilogram of spacecraft mass. [Hint: Write the energy equation for escape from Earth with v' as the velocity, relative to Earth, but far from Earth; then let v' + vQ be the escape velocity from the Sun.]

How long will it take a 1750-W motor to lift a 335-kg piano to a sixth-story window 16.0 m above?

If a car generates 18 hp when traveling at a steady 95 km/h, what must be the average force exerted on the car due to friction and air resistance?

An 85-kg football player traveling 5.0 m/s is stopped in 1.0 s by a tackier, (a) What is the original kinetic energy of the player? (b) What average power is required to stop him?

A driver notices that her 1080-kg car slows down from 95 km/h to 65 km/h in about 7.0 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 80 km/h?

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

An outboard motor for a boat is rated at 55 hp. If it can move a particular boat at a steady speed of 35 km/h, what is the total force resisting the motion of the boat?

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?

During a workout, football players ran up the stadium stairs in 75 s. The stairs are 78 m long and inclined at an angle of 33. If a player has a mass of 92 kg, estimate his average power output on the way up. Ignore friction and air resistance.

A pump lifts 21.0 kg of water per minute through a height of 3.50 m. What minimum output rating (watts) must the pump motor have?

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

A 75-kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a 23 hill. The skier is pulled a distance x = 220 m along the incline and it takes 2.0 min to reach the top of the hill. If the coefficient of kinetic friction between the snow and skis is /% = 0.10, what horsepower engine is required if 30 such skiers (max) are on the rope at one time?

The position of a 280-g object is given (in meters) by x = 5.013 8.012 - 441, where t is in seconds. Determine the net rate of work done on this object (a) at t = 2.0 s and (b) at t = 4.0 s. (c) What is the average net power input during the interval from t = 0 s to t = 2.0 s, and in the interval from t = 2.0 s to 4.0 s?

A bicyclist coasts down a 6.0 hill at a steady speed of 4.0 m/s. Assuming a total mass of 75 kg (bicycle plus rider), what must be the cyclists power output to climb the same hill at the same speed?

Draw a potential energy diagram, U vs. x, and analyze the motion of a mass m resting on a frictionless horizontal table and connected to a horizontal spring with stiffness constant k. The mass is pulled a distance to the right so that the spring is stretched a distance x0 initially, and then the mass is released from rest.

The spring of Problem 75 has a stiffness constant k = 160 N/m. The mass m = 5.0 kg is released from rest when the spring is stretched x0 = 1.0 m from equilibrium. Determine (a) the total energy of the system; (b) the kinetic energy when x = |x 0; (c) the maximum kinetic energy; ((d) the maximum speed and at what positions it

The potential energy of the two atoms in a diatomic (two-atom) molecule can be written where r is the distance between the two atoms and a and b are positive constants, (a) At what values of r is U{r) a minimum? A maximum? (b) At what values of r is U(r) = 0? (c) Plot U(r) as a function of r from r = 0 to r at a value large enough for all the features in (a) and (b) to show, (d) Describe the motion of one atom with respect to the second atom when E < 0, and when E > 0. (e) Let F be the force one atom exerts on the other. For what values of r is F > 0, F < 0, F = 0? ( /) Determine F as a function of r.

The binding energy of a two-particle system is defined as the energy required to separate the two particles from their state of lowest energy to r = oo. Determine the binding energy for the molecule discussed in Problem 77.

What is the average power output of an elevator that lifts 885 kg a vertical height of 32.0 m in 11.0 s?

A projectile is fired at an upward angle of 48.0 from the top of a 135-m-high cliff with a speed of 165 m/s. What will be its speed when it strikes the ground below? (Use conservation of energy.)

Water flows over a dam at the rate of 580 kg/s and falls vertically 88 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 55% efficiency.

A bicyclist of mass 75 kg (including the bicycle) can coast down a 4.0 hill at a steady speed of 12 km/h. Pumping hard, the cyclist can descend the hill at a speed of 32 km/h. Using the same power, at what speed can the cyclist climb the same hill? Assume the force of friction is proportional to the square of the speed v; that is, Ffr = bv2, where b is a constant.

A 62-kg skier starts from rest at the top of a ski jump, point A in Fig. 8-41, and travels down the ramp. If friction and air resistance can be neglected, (a) determine her speed vq when she reaches the horizontal end of the ramp at B. (b) Determine the distance s to where she strikes the ground at C. FIGURE 8-Problems 83 and 84.

Repeat Problem 83, but now assume the ski jump turns upward at point B and gives her a vertical component of velocity (at B) of 3.0 m/s.

A ball is attached to a horizontal cord of length whose other end is fixed, Fig. 8-42. (a) If the ball is released, what will be its speed at the lowest point of its path? (b) A peg is located a distance h directly below the point of attachment of the cord. If h = 0.80, what will be the speed of the ball when it reaches the top of its circular path about the peg? FIGURE 8-42 Problems 85 and 86.

Show that h must be greater than 0.60 if the ball in Fig. 8-42 is to make a complete circle about the peg.

Show that on a roller coaster with a circular vertical loop (Fig. 8-43), the difference in your apparent weight at the top of the loop and the bottom of the loop is 6 gsthat is, six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this answer doesnt depend on the size of the loop or how fast you go through it. FIGURE 8-43 Problem 87.

If you stand on a bathroom scale, the spring inside the scale compresses 0.50 mm, and it tells you your weight is 760 N. Now if you jump on the scale from a height of 1.0 m, what does the scale read at its peak?

A 65-kg hiker climbs to the top of a 4200-m-high mountain. The climb is made in 5.0 h starting at an elevation of 2800 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.

The small mass m sliding without friction along the looped track shown in Fig. 8-44 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. Next, if the actual release height is 2h, calculate the normal force exerted (b) by the track at the bottom of the loop, (c) by the track at the top of the loop, and (d) by the track after the block exits the loop onto the flat section.FIGURE 8-44 Problem 90.

A 56-kg student runs at 5.0 m/s, grabs a hanging rope, and swings out over a lake (Fig. 8-45). He releases the rope when his velocity is zero, (a) What is the angle 0 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? FIGURE 8-45 Problem 91.

The nuclear force between two neutrons in a nucleus is described roughly by the Yukawa potential U{r) = -U 0j e ~ rlr\ where r is the distance between the neutrons and C/0 and r0(~ 10-15m) are constants, (a) Determine the force F(r). (b) What is the ratio F(3r0)/F(r0)? (c) Calculate this same ratio for the force between two electrically charged particles where U(r) = C/r, with C a constant. Why is the Yukawa force referred to as a short-range force?

A fire hose for use in urban areas must be able to shoot a stream of water to a maximum height of 33 m. The water leaves the hose at ground level in a circular stream 3.0 cm in diameter. What minimum power is required to create such a stream of water? Every cubic meter of water has a mass of 1.00 X 103kg.

A 16-kg sled starts up a 28 incline with a speed of 2.4 m/s. The coefficient of kinetic friction is /z^ = 0.25. (a) How far up the incline does the sled travel? (b) What condition must you put on the coefficient of static friction if the sled is not to get stuck at the point determined in part (a)l (c) If the sled slides back down, what is its speed when it returns to its starting point?

The Lunar Module could make a safe landing if its vertical velocity at impact is 3.0 m/s or less. Suppose that you want to determine the greatest height h at which the pilot could shut off the engine if the velocity of the lander relative to the surface is (a) zero; (b) 2.0 m/s downward; (c) 2.0 m/s upward. Use conservation of energy to determine h in each case. The acceleration due to gravity at the surface of the Moon is 1.62 m/s2.

Proper design of automobile braking systems must account for heat buildup under heavy braking. Calculate the thermal energy dissipated from brakes in a 1500-kg car that descends a 17 hill. The car begins braking when its speed is 95 km/h and slows to a speed of 35 km/h in a distance of 0.30 km measured along the road.

Some electric power companies use water to store energy. Water is pumped by reversible turbine pumps from a low reservoir to a high reservoir. To store the energy produced in 1.0 hour by a 180-MW 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 380 m above the lower one, and we can neglect the small change in depths of each. Water has a mass of 1.00 X 103 kg for every 1.0 m3.

Estimate the energy required from fuel to launch a 1465-kg satellite into orbit 1375 km above the Earths surface. Consider two cases: (a) the satellite is launched into an equatorial orbit from a point on the Earths equator, and (b) it is launched from the North Pole into a polar orbit.

A satellite is in an elliptic orbit around the Earth (Fig. 8-46). Its speed at the perigee A is 8650 m/s. (a) Use conservation of energy to determine its speed at B. The radius of the Earth is 6380 km. (b) Use conservation of energy to determine the speed at the apogee C. FIGURE 8-46 Problem 99.

Suppose the gravitational potential energy of an object of mass m at a distance r from the center of the Earth is given by U(r) = - GMm where a is a positive constant and e is the exponential function. (Newtons law of universal gravitation has a = 0 ). (a) What would be the force on the object as a function of rl (b) What would be the objects escape velocity in terms of the Earths radius RE7

(a) If the human body could convert a candy bar directly into work, how high could a 76-kg man climb a ladder if he were fueled by one bar (= 1100 kJ)? (ft) If the man then jumped off the ladder, what will be his speed when he reaches the bottom?

Electric energy units are often expressed in the form of kilowatt- hours. (a) Show that one kilowatt-hour (kWh) is equal to 3.6 X 106J. (b) If a typical family of four uses electric energy at an average rate of 580 W, how many kWh would their electric bill show 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?

Electric energy units are often expressed in the form of kilowatt- hours. (a) Show that one kilowatt-hour (kWh) is equal to 3.6 X 106J. (b) If a typical family of four uses electric energy at an average rate of 580 W, how many kWh would their electric bill show 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?FIGURE 8-47 Problem 103. (a) Bungee jumper about to jump, (b) Bungee cord at its unstretched length, (c) Maximum stretch of cord.

In a common test for cardiac function (the stress test), the patient walks on an inclined treadmill (Fig. 8-48). Estimate the power required from a 75-kg patient when the treadmill is sloping at an angle of 12 and the velocity is 3.3 km/h. (How does this power compare to the power rating of a lightbulb?)FIGURE 8-48 Problem 104.

(a) If a volcano spews a 450-kg rock vertically upward a distance of 320 m, what was its velocity when it left the volcano? (b) If the volcano spews 1000 rocks of this size every minute, estimate its power output.

A film of Jesse Owenss famous long jump (Fig. 8-49) in the 1936 Olympics shows that his center of mass rose 1.1m from launch point to the top of the arc. What minimum speed did he need at launch if he was traveling at 6.5 m/s at the top of the arc?FIGURE 8-49 Problem 106.

An elevator cable breaks when a 920-kg elevator is 24 m above a huge spring (k = 2.2 X 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).

A particle moves where its potential energy is given by U(r) = U0 [(2/r2) - (1 /r)\. (a) Plot U(r) versus r. Where does the curve cross the U(r) = 0 axis? At what value of r does the minimum value of U(r) occur? (b) Suppose that the particle has an energy of E = -0.050C/0. Sketch in the approximate turning points of the motion of the particle on your diagram. What is the maximum kinetic energy of the particle, and for what value of r does this occur?

A particle of mass m moves under the influence of a potential energy U(x) = + bx where a and b are positive constants and the particle is restricted to the region x > 0. Find a point of equilibrium for the particle and demonstrate that it is stable.

The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or F(r) = F0 [2(cr/r)13 (cr/r)7], where r is the separation between the two atoms, and a and F0 are constant. For an oxygen molecule (which is diatomic) F0 = 9.60 X 10-11 N and a = 3.50 X 10-11 m. (a) Integrate the equation for F(r) to determine the potential energy U(r) of the oxygen molecule. (b) Find the equilibrium distance r0 between the two atoms, (c) Graph F(r) and U(r) between 0.9 r0 and 2.5 r0.