A particle moves in a vertical plane along the closed path

The spring in Figure CP11.73 has a spring constant of 1000 N/m. It is compressed 15 cm, then launches a 200 g block. The horizontal surface is frictionless, but the block’s coefficient of kinetic friction on the incline is 0.20. What distance \(d\) does the block sail through the air? ________________ Equation Transcription: Text Transcription: d

Problem 72CP A 10.2 kg weather rocket generates a thrust of 200 N. The rocket, pointing upward, is clamped to the top of a vertical spring. The bottom of the spring, whose spring constant is 500 N/m, is anchored to the ground. a. Initially, before the engine is ignited, the rocket sits at rest on top of the spring. How much is the spring compressed? ________________ b. After the engine is ignited, what is the rocket’s speed when the spring has stretched 40 cm? For comparison, what would be the rocket’s speed after traveling this distance if it weren’t attached to the spring?

Problem 75CP A gardener pushes a 12 kg lawnmower whose handle is tilted up 37° above horizontal. The lawnmower’s coefficient of rolling friction is 0.15. How much power does the gardener have to supply to push the lawnmower at a constant speed of 1.2 m/s? Assume his push is parallel to the handle.

The equation \(m g y\) for gravitational potential energy is valid only for objects near the surface of a planet. Consider two very large objects of mass \(m_{1}\) and \(m_{2}\), such as stars or planets, whose centers are separated by the large distance \(r\). These two large objects exert gravitational forces on each other. You'll learn in Chapter 13 that the gravitational potential energy is \(U=-\frac{G m_{1} m_{2}}{r}\) where \(G=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} / \mathrm{kg}^{2}\) is the gravitational constant. a. Sketch a graph of \(U\) versus \(r\). The mathematical difficulty at \(r=0\) is not a physically significant difficulty because the masses will collide before they get that close together. b. What separation \(r\) has been chosen as the point of zero potential energy? Does this make sense? Explain. c. Two stars are at rest \(1.0 \times 10^{14} \mathrm{~m}\) apart. This is about 10 times the diameter of the solar system. The first star is the size of our sun, with a mass of \(2.0 \times 10^{30} \mathrm{~kg}\) and a radius of \(7.0 \times 10^{8} \mathrm{~m}\). The second star has mass \(8.0 \times 10^{30} \mathrm{~kg}\) and radius of \(11.0 \times 10^{8} \mathrm{~m}\). Gravitational forces pull the two stars together. What is the speed of each star at the moment of impact? ________________ Equation Transcription: Text Transcription: mgy m_{1} m_{2} r U = -frac{G m_{1} m_{2}}{r} G = 6.67 X 10^{-11} Nm^{2} / kg^2 U r r = 0 r 1.0 X 10^{14} m 2.0 X 10^{30} kg 7.0 X 10^{8} m 8.0 Xs 10^{30} kg 11.0 X 10^{8} m

A process occurs in which a system's potential energy decreases while the system does work on the environment. Does the system's kinetic energy increase, decrease, or stay the same? Or is there not enough information to tell? Explain.

Evaluate the dot product \(\vec{A} \cdot \vec{B}\) if a. \(\vec{A}=3 \hat{\imath}+4 \hat{\jmath}\) and \(\vec{B}=2 \hat{\imath}-6 \hat{\jmath}\). b. \(\vec{A}=3 \hat{\imath}-2 \hat{\jmath}\) and \(\vec{B}=6 \hat{\imath}+4 \hat{\jmath}\) ________________ Equation Transcription: and and Text Transcription: vec{A} cdot vec{B} vec{A} = 3 hat i + 4 hat j and vec{B} = 2 hat i - 6 hat j vec{A} = 3 hat i - 2 hat j and vec{B} = 6 hat i + 4 hat j

Problem 2CQ A process occurs in which a system’s potential energy increases while the environment does work on the system. Does the system’s kinetic energy increase, decrease, or stay the same? Or is there not enough information to tell? Explain.

Problem 4CQ You drop a ball from a high balcony and it falls freely. Does the ball’s kinetic energy increase by equal amounts in equal time intervals, or by equal amounts in equal distances? Explain.

Evaluate the dot product \(\vec{A} \cdot \vec{B}\) if a. \(\vec{A}=4 \hat{\imath}-2 \hat{\jmath}\) and \(\vec{B}=-2 \hat{\imath}-3 \hat{\jmath}\) b. \(\vec{A}=-4 \hat{\imath}+2 \hat{\jmath}\) and \(\vec{B}=2 \hat{\imath}+4 \hat{\jmath}\) Equation Transcription: Text Transcription: vec A cdot vec B vec A = 4 \hat i - 2 \hat j vec B = -2 \hat i - 3 \hat j vec A = -4 \hat i + 2 \hat j vec B = 2 \hat i + 4 \hat j

What is the angle \(\theta\) between vectors \(\vec{A}\) and \(\vec{B}\) in each part of Exercise 1? ________________ Equation Transcription: Text Transcription: theta vec{A} vec{B}

What is the angle \(\theta\) between vectors \(\vec{A}\) and \(\vec{B}\) in each part of Exercise 2? ________________ Equation Transcription: Text Transcription: theta vec{A} vec{B}

Problem 3CQ The kinetic energy of a system decreases while its potential energy and thermal energy are unchanged. Does the environment do work on the system, or does the system do work on the environment? Explain.

A particle moves in a vertical plane along the closed path seen in Figure Q11.5, starting at A and eventually returning to its starting point. How much work is done on the particle by gravity? Explain.

Evaluate the dot product of the three pairs of vectors in Figure EX11.5.

Evaluate the dot product of the three pairs of vectors in Figure EX11.6.

Problem 6CQ A 0.2 kg plastic cart and a 20 kg lead cart both roll without friction on a horizontal surface. Equal forces are used to push both carts forward a distance of 1 m, starting from rest. After traveling 1 m, is the kinetic energy of the plastic cart greater than, less than, or equal to the kinetic energy of the lead cart? Explain.

Problem 8CQ a. If the force on a particle at some point in space is zero, must its potential energy also be zero at that point? Explain. ________________ b. If the potential energy of a particle at some point in space is zero, must the force on it also be zero at that point? Explain.

How much work is done by the force \(\vec{F}=(-3.0 \hat{\imath}+6.0 \hat{\jmath}) \mathrm{N}\) on a particle that moves through displacement (a) \(\Delta \vec{r}=2.0 \hat{\imath} \mathrm{m}\) and \((\mathrm{b}) \Delta \vec{r}=2.0 \hat{\jmath} \mathrm{m}\) ? ________________ Equation Transcription: Text Transcription: vec{F} = (-3.0 hat{i} + 6.0 hat{j}) N Delta vec{r} = 2.0 hat{i} m Delta vec{r} = 2.0 hat{j} m

Problem 7CQ You need to raise a heavy block by pulling it with a massless rope. You can either (a) pull the block straight up height h, or (b) pull it up a long, frictionless plane inclined at a 15° angle until its height has increased by h. Assume you will move the block at constant speed either way. Will you do more work in case a or case b? Or is the work the same in both cases? Explain.

How much work is done by the force \(\vec{F}=(-4.0 \hat{\imath}-6.0 \hat{\jmath}) \mathrm{N}\) on a particle that moves through displacement (a) \(\Delta \vec{r}=-3.0 \hat{\imath} \mathrm{m}\) and \((b) \Delta \vec{r}=(3.0 \hat{\imath}-2.0 \hat{\jmath}) \mathrm{m} ?\) ________________ Equation Transcription: Text Transcription: vec{F} = (-4.0 hat{i} - 6.0 hat{j}) N Delta vec{r} = -3.0 hat{i} m Delta vec{r} = 3.0 hat{i} -2.0 hat{j} m

Problem 9CQ A car traveling at 60 mph slams on its brakes and skids to a halt. What happened to the kinetic energy the car had just before stopping?

Problem 9E A 20 g plastic ball is moving to the left at 30 m/s. How much work must be done on the ball to cause it to move to the right at 30 m/s?

Problem 10E A 2.0 kg book is lying on a 0.75-m-high table. You pick it up and place it on a bookshelf 2.3 m above the floor. During this process, a. How much work does gravity do on the book? b. How much work does your hand do on the book?

The two ropes seen in Figure EX11.11 are used to lower a \(255 kg\) piano \(5.00 m\) from a second-story window to the ground. How much work is done by each of the three forces? ________________ Equation Transcription: Text Transcription: 255 kg 5.00 m

Problem 12CQ The motor of a crane uses power P to lift a steel beam. By what factor must the motor’s power increase to lift the beam twice as high in half the time?

Problem 10CQ What energy transformations occur as a skier glides down a gentle slope at constant speed?

The three ropes shown in the bird’s-eye view of Figure EX11.12 are used to drag a crate \(3.0 \ m\) across the floor. How much work is done by each of the three forces? Equation Transcription: Text Transcription: 3.0 m

Give a specific example of a situation in which a. \(W_{\mathrm{ext}} \rightarrow K\) with \(\Delta U=0\) and \(\Delta E_{\mathrm{th}}=0\). b. \(W_{\mathrm{ext}} \rightarrow E_{\mathrm{th}}\) with \(\Delta K=0\) and \(\Delta U=0\). ________________ Equation Transcription: with and with and Text Transcription: W_{\mathrm{ext}} \rightarrow K with \Delta U=0 and \Delta E_{\mathrm{th}}=0 W_{\mathrm{ext}} \rightarrow E_{\mathrm{th}} with \Delta K=0 and \Delta U=0

One measure of the intensity of a transition of frequency v is the oscillator strength, f, which is defined as \(f=\frac{8 \pi^2 m_{\mathrm{e}} v\left|\boldsymbol{\mu}_{\mathrm{f}}\right|^2}{3 h e^2}\) Consider an electron in an atom to be oscillating harmonically in one dimension (the three-dimensional version of this model was used in early attempts to describe atomic structure). The wavefunctions for such an electron are those in Table 9.1. Show that the oscillator strength for the transition of this electron from its ground state is exactly \(\frac{1}{3}\).

Figure EX11.14 is the force-versus-position graph for a particle moving along the x-axis. Determine the work done on the particle during each of the three intervals \(0–1 m, 1–2 m,\) and \(2–3 m\). ________________ Equation Transcription: Text Transcription: 0-1 m, 1 -2 m 2 -3 m

A \(500 g\) particle moving along the\( x\)-axis experiences the force shown in Figure EX11.15. The particle’s velocity is \(2.0 m/s\) at \(x = 0 m\). What is its velocity at \(x = 1 m, 2 m\), and \(3 m\)? ________________ Equation Transcription: Text Transcription: 500 g x 2.0 m/s x = 0 m x = 1m, 2m 3m

A particle has the potential energy shown in Figure EX11.18. What is the \(x\)-component of the force on the particle at \(x = 5, 15, 25\), and \(35 cm\)? ________________ Equation Transcription: Text Transcription: x x = 5, 15, 25 35 cm

A particle has the potential energy shown in Figure EX11.19. What is the \(y\)-component of the force on the particle at \(y = 0.5 m\) and \(4 m\)? ________________ Equation Transcription: Text Transcription: y y = 0.5 m 4 m

A \(2.0 kg\) particle moving along the \(x\)-axis experiences the force shown in Figure EX11.16. The particle’s velocity is \(4.0 m/s\) at \(x = 0 m\). What is its velocity at \(x = 2 m\) and \(4 m\)? ________________ Equation Transcription: Text Transcription: 2.0 kg x 4.0 m/s x = 0 m x = 2 m 4 m

Problem 20E A particle moving along the y-axis has the potential energy U = 4y3 J, where y is in m. What is the y-component of the force on the particle at y =0 m, 1 m, and 2 m?

Problem 21E A particle moving along the x-axis has the potential energy U =10/x J, where x is in m. What is the x-component of the force on the particle at x = 2 m, 5 m, and 8 m?

A \(500 \mathrm{~g}\) particle moving along the \(x\)-axis experiences the force shown in FIGURE EX11.17. The particle goes from \(v_{x}=2.0 \mathrm{~m} / \mathrm{s}\) at \(x=0 \mathrm{~m}\) to \(v_{x}=6.0 \mathrm{~m} / \mathrm{s}\) at \(x=2 \mathrm{~m}\). What is \(F_{\max }\) ? ________________ Equation Transcription: Text Transcription: 500 g x v_{x} = 2.0 m/s x = 0 m v_{x} = 6.0 m/s x = 2 m F_{max }

Problem 22E The mass of a carbon atom is 2.0 × 10?26 kg. a. What is the kinetic energy of a carbon atom moving with a speed of 500 m/s? ________________ b. Two carbon atoms are joined by a spring-like carbon-carbon bond. The potential energy stored in the bond has the value you calculated in part a if the bond is stretched 0.050 nm. What is the bond’s spring constant?

Problem 23E In Part IV you’ll learn to calculate that 1 mole (6.02 × 1023 atoms) of helium atoms in the gas phase has 3700 J of microscopic kinetic energy at room temperature. If we assume that all atoms move with the same speed, what is that speed? The mass of a helium atom is 6.68 × 10?27 kg.

Problem 25E A system loses 400 J of potential energy. In the process, it does 400 J of work on the environment and the thermal energy increases by 100 J. Show this process on an energy bar chart.

Problem 24E A 20 kg child slides down a 3.0-m-high playground slide. She starts from rest, and her speed at the bottom is 2.0 m/s. a. Describe the energy transfers and transformations occurring during the slide. ________________ b. What is the change in the combined thermal energy of the slide and the seat of her pants?

Problem 26E A system loses 500 J of kinetic energy while gaining 200 J of potential energy. The thermal energy increases 100 J. Show this process on an energy bar chart.

How much work is done by the environment in the process shown in Figure EX11.27? Is energy transferred from the environment to the system or from the system to the environment?

Problem 32E Which consumes more energy, a 1.2 kW hair dryer used,for 10 min or a 10 W night light left on for 24 h?

Problem 31E At midday, solar energy strikes the earth with an intensity of about 1 kW/m2. What is the area of a solar collector that could collect 150 MJ of energy in 1 h? This is roughly the energy content of 1 gallon of gasoline.

Problem 28E A cable with 20.0 N of tension pulls straight up on a 1.02 kg block that is initially at rest. What is the block’s speed after being lifted 2.00 m? Solve this problem using work and energy.

Problem 33E A 2.0 hp electric motor on a water well pumps water from 10 m below the surface. The density of water is 1.0 kg per liter. How many liters of water does the motor pump in 1 h?

Problem 30E a. How much work must you do to push a 10 kg block of steel across a steel table at a steady speed of 1.0 m/s for 3.0 s? ________________ b. What is your power output while doing so?

Problem 29E a. How much work does an elevator motor do to lift a 1000 kg elevator a height of 100 m at a constant speed? b. How much power must the motor supply to do this in 50 s at constant speed?

Problem 34E A 50 kg sprinter, starting from rest, runs 50 m in 7.0 s at constant acceleration. a. What is the magnitude of the horizontal force acting on the sprinter? ________________ b. What is the sprinter’s power output at 2.0 s, 4.0 s, and 6.0 s?

Problem 35E a. Estimate the height in meters of the two flights of stairs that go from the first to the third floor of a building. ________________ b. Estimate how long it takes you to run up these two flights of stairs. ________________ c. Estimate your power output in both watts and horsepower while running up the stairs.

Problem 36E A 70 kg human sprinter can accelerate from rest to 10 m/s in 3.0 s. During the same time interval, a 30 kg greyhound can go from rest to 20 m/s. What is the average power output of each? Average power over a time interval ?t is ?E/ ?t.

A particle moves from A to D in Figure P11.37 while experiencing force \(\vec{F}=(6 \hat{\imath}+8 \hat{\jmath}) \mathrm{N}.\) How much work does the force do if the particle follows path (a) ABD, (b) ACD, and (c) AD? Is this a conservative force? Explain. ________________ Equation Transcription: Text Transcription: vec{F} = (6 hat{i} + 8 hat{j}) N

Figure P11.41 is the velocity-versus-time graph of a \(500 g\) particle that starts at \(x = 0 m\) and moves along the \(x\)-axis. Draw graphs of the following by calculating and plotting numerical values at \(t\) = 0, 1, 2, 3, and 4 s. Then sketch lines or curves of the appropriate shape between the points. Make sure you include appropriate scales on both axes of each graph. a. Acceleration versus time. b. Position versus time. c. Kinetic energy versus time. d. Force versus time. e. Use your \(F_{x}\)-versus- \(t\) graph to determine the impulse delivered to the particle during the time interval 0-2 s and also the interval 2-4 s. f. Use the impulse-momentum theorem to determine the particle’s velocity at \(t = 2 s\) and at \(t = 4 \). Do your results agree with the velocity graph? g. Now draw a graph of force versus position. This requires no calculations; just think carefully about what you learned in parts a to d. h. Use your \(F_{x}\)-versus- \(x\)graph to determine the work done on the particle during the time interval 0-2 s and also the interval 2-4 s. i. Use the work-kinetic energy theorem to determine the particle’s velocity at \(t = 2 s\) and at \(t = 4 s\). Do your results agree with the velocity graph? ________________ Equation Transcription: Text Transcription: 500 g x = 0 x t F_x t t = 2 s t = 4 s F_x x t = 2 s t = 4 s

A 100 g particle experiences the one-dimensional, conservative force \(F_{x}\) shown in FIGURE P11.38. a. Draw a graph of the potential energy \(U\) from \(x=0 \mathrm{~m}\) to \(x=5 \mathrm{~m}\). Let the zero of the potential energy be at \(x=0 \mathrm{~m}\). Hint: Think about the definition of potential energy and the geometric interpretation of the work done by a varying force. b. The particle is shot toward the right from \(x=1.0 \mathrm{~m}\) with a speed of 25 m/s. What is the particle's mechanical energy? c. Draw the particle's total energy line on your graph of part a. d. Where is the particle's turning point? ________________ Equation Transcription: Text Transcription: F_x U x = 0 m x = 5 m x = 0 m x = 1.0 m

a. FIGURE P11.40a shows the force \(F_{x}\) exerted on a particle that moves along the \(x\)-axis. Draw a graph of the particle's potential energy as a function of position \(x\). Let \(U\) be zero at \(x=0 \mathrm{~m}\). b. FIGURE P11.40b shows the potential energy \(U\) of a particle that moves along the \(x\)-axis. Draw a graph of the force \(F_{x}\) as a function of position \(x\). ________________ Equation Transcription: Text Transcription: F_x x x U x = 0 m U x F_x x

A 10 g particle has the potential energy shown in Figure P11.39. a. Draw a force-versus-position graph from\(x = 0 cm\) to\(x = 8 cm\). b. How much work does the force do as the particle moves from \(x = 2 cm\) to \(x = 6 cm\)? c. What speed does the particle need at \(x = 2 cm\) to arrive at \(x = 6 cm\) with a speed of 10 m/s? ________________ Equation Transcription: Text Transcription: x = 0 cm x = 8 cm x = 2 cm x = 6 cm x = 2 cm x = 6 cm

Problem 42P A 1000 kg elevator accelerates upward at 1.0 m/s2 for 10 m, starting from rest. a. How much work does gravity do on the elevator? ________________ b. How much work does the tension in the elevator cable do on the elevator? ________________ c. Use the work-kinetic energy theorem to find the kinetic energy of the elevator as it reaches 10 m. ________________ d. What is the speed of the elevator as it reaches 10 m?

Problem 43P Bob can throw a 500 g rock with a speed of 30 m/s. He moves his hand forward 1.0 m while doing so. a. How much work does Bob do on the rock? ________________ b. How much force, assumed to be constant, does Bob apply to the rock? ________________ c. What is Bob’s maximum power output as he throws the rock?

Problem 46P Susan’s 10 kg baby brother Paul sits on a mat. Susan pulls the mat across the floor using a rope that is angled 30° above the floor. The tension is a constant 30 N and the coefficient of friction is 0.20. Use work and energy to find Paul’s speed after being pulled 3.0 m.

Problem 47P A horizontal spring with spring constant 100 N/m is compressed 20 cm and used to launch a 2.5 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Use work and energy to find how far the box slides across the rough surface before stopping.

Problem 45P Sam, whose mass is 75 kg, straps on his skis and starts down a 50-m-high, 20° frictionless slope. A strong headwind exerts a horizontal force of 200 N on him as he skies. Use work and energy to find Sam’s speed at the bottom.

Problem 44P a. Starting from rest, a crate of mass m is pushed up a frictionless slope of angle ? by a horizontal force of magnitude F. Use work and energy to find an expression for the crate’s speed v when it is at height h above the bottom of the slope. ________________ b. Doug uses a 25 N horizontal force to push a 5.0 kg crate up a 2.0-m-high, 20° frictionless slope. What is the speed of the crate at the top of the slope?

Problem 48P a. A box of mass m and initial speed v0 slides distance d across a horizontal floor before coming to rest. Use work and energy to find an expression for the coefficient of kinetic friction. ________________ b. A baggage handler throws a 15 kg suitcase along the floor of an airplane luggage compartment with a speed of 1.2 m/s. The suitcase slides 2.0 m before stopping. What is the suitcase’s coefficient of kinetic friction on the floor?

Problem 49P Truck brakes can fail if they get too hot. In some mountainous areas, ramps of loose gravel are constructed to stop runaway trucks that have lost their brakes. The combination of a slight upward slope and a large coefficient of rolling resistance as the truck tires sink into the gravel brings the truck safely to a halt. Suppose a gravel ramp slopes upward at 6.0° and the coefficient of rolling friction is 0.40. Use work and energy to find the length of a ramp that will stop a 15,000 kg truck that enters the ramp at 35 m/s (? 75 mph).

A freight company uses a compressed spring to shoot \(2.0 kg\) packages up a \(1.0-m\)-high frictionless ramp into a truck, as Figure P11.50 shows. The spring constant is 500 N/m and the spring is compressed \(30 cm\). a. What is the speed of the package when it reaches the truck? b. A careless worker spills his soda on the ramp. This creates a \(50-cm\)-long sticky spot with a coefficient of kinetic friction 0.30. Will the next package make it into the truck? ________________ Equation Transcription: Text Transcription: 2.0 kg 1.0-m 30 cm 50- cm

Use work and energy to find an expression for the speed of the block in Figure P11.51 just before it hits the floor if (a) the coefficient of kinetic friction for the block on the table is \(\mu_{\mathrm{k}}\) and (b) the table is frictionless. ________________ Equation Transcription: Text Transcription: mu_k

Problem 52P An 8.0 kg crate is pulled 5.0 m up a 30° incline by a rope angled 18° above the incline. The tension in the rope is 120 N, and the crate’s coefficient of kinetic friction on the incline is 0.25. a. How much work is done by tension, by gravity, and by the normal force? ________________ b. What is the increase in thermal energy of the crate and incline?

You’ve taken a summer job at a water park. In one stunt, a water skier is going to glide up the \(2.0-m\)-high frictionless ramp shown in Figure P11.53, then sail over a \(5.0-m\)-wide tank filled with hungry sharks. You will be driving the boat that pulls her to the ramp. She’ll drop the tow rope at the base of the ramp just as you veer away. What minimum speed must you have as you reach the ramp in order for her to live to do this again tomorrow? ________________ Equation Transcription: Text Transcription: 20-m 5.0-m

Problem 55P A 50 g ice cube can slide without friction up and down a 30° slope. The ice cube is pressed against a spring at the bottom of the slope, compressing the spring 10 cm. The spring constant is 25 N/m. When the ice cube is released, what distance will it travel up the slope before reversing direction?

Problem 54P A 50 kg ice skater is gliding along the ice, heading due north at 4.0 m/s. The ice has a small coefficient of static friction, to prevent the skater from slipping sideways, but ?k = 0. Suddenly, a wind from the northeast exerts a force of 4.0 N on the skater. a. Use work and energy to find the skater’s speed after gliding 100 min this wind. ________________ b. What is the minimum value of ?s that allows her to continue moving straight north?

The spring shown in Figure P11.57 is compressed \(50 cm\) and used to launch a \(100 kg\) physics student. The track is frictionless until it starts up the incline. The student’s coefficient of kinetic friction on the \(30^{\circ}\) incline is 0.15. a. What is the student’s speed just after losing contact with the spring? b. How far up the incline does the student go? ________________ Equation Transcription: Text Transcription: 50 cm 100 kg 30^circ

Problem 56P A 5.0 kg box slides down a 5.0-m-high frictionless hill, starting from rest, across a 2.0-m-wide horizontal surface, then hits a horizontal spring with spring constant 500 N/m. The other end of the spring is anchored against a wall. The ground under the spring is frictionless, but the 2.0-m-wide horizontal surface is rough. The coefficient of kinetic friction of the box on this surface is 0.25. a. What is the speed of the box just before reaching the rough surface? ________________ b. What is the speed of the box just before hitting the spring? ________________ c. How far is the spring compressed? ________________ d. Including the first crossing, how many complete trips will the box make across the rough surface before coming to rest?

Problem 59P Show that Hooke’s law for an ideal spring is a conservative force. To do so, first calculate the work done by the spring as it expands from A to B. Then calculate the work done by the spring as it expands from A to point C, which is beyond B, then returns from C to B.

Problem 58P A block of mass m starts from rest at height h. It slides down a frictionless incline, across a rough horizontal surface of length L, then up a frictionless incline. The coefficient of kinetic friction on the rough surface is ?k. a. What is the block’s speed at the bottom of the first incline? ________________ b. How high does the block go on the second incline?

Problem 60P A clever engineer designs a “sprong” that obeys the force law Fx =?q(x ? xe)3, where xe is the equilibrium position of the end of the sprong and q is the sprong constant. For simplicity, we’ll let xe = 0 m. Then Fx = ? qx3. a. What are the units of q ? ________________ b. Find an expression for the potential energy of a stretched or compressed sprong. ________________ c. A sprong-loaded toy gun shoots a 20 g plastic ball. What is the launch speed if the sprong constant is 40,000, with the units you found in part a, and the sprong is compressed 10 cm? Assume the barrel is frictionless.

Problem 61P A particle of mass m starts from x0 = 0 m with v0 > 0 m/s. The particle experiences the variable force Fx =F0sn(cx)as it moves to the right along the x-axis, where F0 and c are constants. a. What are the units of F0? ________________ b. What are the units of c? ________________ c. At what position xmax does the force first reach a maximum value? Your answer will be in terms of the constants F0 and c and perhaps other numerical constants. ________________ d. Sketch a graph of F versus x from x0 to xmax. ________________ e. What is the particle’s velocity as it reaches xmax? Give your answer in terms of m, v0, F0,and c.

Problem 62P A 5.0 kg cat leaps from the floor to the top of a 95-cm-high table. If the cat pushes against the floor for 0.20 s to accomplish this feat, what was her average power output during the pushoff period?

Problem 64P When you ride a bicycle at constant speed, nearly all the energy you expend goes into the work you do against the drag force of the air. Model a cyclist as having cross-section area 0.45 m2and, because the human body is not aerodynamically shaped, a drag coefficient of 0.90. a. What is the cyclist’s power output while riding at a steady 7.3 m/s (16 mph)? ________________ b. Metabolic power is the rate at which your body “burns” fuel to power your activities. For many activities, your body is roughly 25% efficient at converting the chemical energy of food into mechanical energy. What is the cyclist’s metabolic power while cycling at 7.3 m/s? ________________ c. The food calorie is equivalent to 4190 J. How many calories does the cyclist burn if he rides over level ground at 7.3 m/s for 1 h?

Problem 63P The human heart pumps the average adult’s 6.0 L (6000 cm3) of blood through the body every minute. The heart must do work to overcome frictional forces that resist blood flow. The average adult blood pressure is 1.3 × 104 N/m2. a. How much work does the heart do to move the 6.0 L of blood completely through the body? ________________ b. What power output must the heart have to do this task once a minute? Hint: When the heart contracts, it applies force to the blood. Pressure is force/area. Model the circulatory system as a single closed tube, with cross-section area A and volume V =6.0 L, filled with blood to which the heart applies a force.

Problem 66P The force required to tow a water skier at speed v is proportional to the speed. That is, Ftow = Av, where A is a proportionality constant. If a speed of 2.5 mph requires 2 hp, how much power is required to tow a water skier at 7.5 mph?

Problem 67P A Porsche 944 Turbo has a rated engine power of 217 hp. 30% of the power is lost in the drive train, and 70% reaches the wheels. The total mass of the car and driver is 1480 kg, and two-thirds of the weight is over the drive wheels. a. What is the maximum acceleration of the Porsche on a concrete surface where µs = 1.00? Hint: What force pushes the car forward? ________________ b. If the Porsche accelerates at amax, what is its speed when it reaches maximum power output? ________________ c. How long does it take the Porsche to reach the maximum power output?

In Problems 68 through 71 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Draw a pictorial representation. c. Finish the solution of the problem. \(\frac{1}{2}(2.0 \mathrm{~kg})(4.0 \mathrm{~m} / \mathrm{s})^{2}+0\) \(+(0.15)(2.0 \mathrm{~kg})\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right)(2.0 \mathrm{~m})=0+0+T(2.0 \mathrm{~m})\) ________________ Equation Transcription: Text Transcription: frac{1}{2}(2.0 kg)(4.0 m / s)^{2} + 0 +(0.15)(2.0 kg) (9.8 m / s^{2})(2.0 m) = 0 + 0 + T(2.0 m)

Problem 65P In a hydroelectric dam, water falls 25 m and then spins a turbine to generate electricity. a. What is ?U of 1.0 kg of water? ________________ b. Suppose the dam is 80% efficient at converting the water’s potential energy to electrical energy. How many kilograms of water must pass through the turbines each second to generate 50 MW of electricity? This is a typical value for a small hydroelectric dam.

In Problems 68 through 71 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Draw a pictorial representation. c. Finish the solution of the problem. \(\frac{1}{2}(20 \mathrm{~kg}) v_{1}^{2}+0\) \(\begin{aligned}&+(0.15)(20 \mathrm{~kg})\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right) \cos 40^{\circ}\left((2.5 \mathrm{~m}) / \sin 40^{\circ}\right) \\&=0+(20 \mathrm{~kg})\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right)(2.5 \mathrm{~m})+0\end{aligned}\) ________________ Equation Transcription: Text Transcription: frac{1}{2}(20 kg) v_{1}^{2} + 0 +(0.15)(20 kg)(9.8 m} / s^2) cos 40^{circ}((2.5 m) / sin 40^{\circ}) = 0 + (20 kg) (9.8 m/ s^2)(2.5 m}) +0

Problem 71P In Problem you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). ________________ b. Draw a pictorial representation. ________________ c. Finish the solution of the problem. T ? (1500 kg)(9.8 m/s2) = (1500kg)(1.0 m/s2) P =T(2.0 m/s)

Problem 70P In Problem you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). ________________ b. Draw a pictorial representation. ________________ c. Finish the solution of the problem. Fpush ? (0.20) (30 kg) (9.8 m/s2) = 0 75W=Fpushv