Solved: For small stretches, the Achilles tendon can be modeled as an ideal spring

Figure 7.14 shows force vectors at different points in space for two forces. Which is conservative and which nonconservative? Explain. (a) (b) Figure 7.14 For Thought and Discussion 1; Problem 30

Is the conservation-of-mechanical-energy principle related to Newtons laws, or is it an entirely separate physical principle? Discuss.

Why cant we define a potential energy associated with friction?

Can potential energy be negative? Can kinetic energy? Can total mechanical energy? Explain.

If the potential energy is zero at a given point, must the force also be zero at that point? Give an example.

If the force is zero at a given point, must the potential energy also be zero at that point? Give an example.

If the difference in potential energy between two points is zero, does that necessarily mean that an object moving between those points experiences no force?

A tightrope walker follows an essentially horizontal rope between two mountain peaks of equal altitude. A climber descends from one peak and climbs the other. Compare the work done by the gravitational force on the tightrope walker and the climber.

If conservation of energy is a law of nature, why do we have programslike mileage requirements for cars or insulation standards for buildingsdesigned to encourage energy conservation?

Determine the work you would have to do to move a block of mass m from point 1 to point 2 at constant speed over the two paths shown in Fig. 7.15. The coefficient of friction has the constant value m over the surface. Note: The diagram lies in a horizontal plane. (b) (a) 2 L L 1 Figure 7.15 Exercises 10 and 11

Now take Fig. 7.15 to lie in a vertical plane, and find the work done by the gravitational force as an object moves from point 1 to point 2 over each of the paths shown.

Rework Example 7.1, now taking the zero of potential energy at street level.

Find the potential energy associated with a 70-kg hiker (a) atop New Hampshires Mount Washington, 1900 m above sea level, and (b) in Death Valley, California, 86 m below sea level. Take the zero of potential energy at sea level.

You fly from Bostons Logan Airport, at sea level, to Denver, altitude 1.6 km. Taking your mass as 65 kg and the zero of potential energy at Boston, whats the gravitational potential energy when youre (a) at the planes 11-km cruising altitude and (b) in Denver?

The potential energy associated with a 60-kg hiker ascending 1250-m-high Camels Hump mountain in Vermont is -240 kJ; the zero of potential energy is taken at the mountaintop. Whats her altitude?

How much energy can be stored in a spring with k = 320 N/m if the maximum allowed stretch is 18 cm?

How far would you have to stretch a spring with k = 1.4 kN/m for it to store 210 J of energy?

A biophysicist grabs the ends of a DNA strand with optical tweezers and stretches it 26 m. How much energy is stored in the stretched molecule if its spring constant is 0.046 pN/m?

A skier starts down a frictionless 32 slope. After a vertical drop of 25 m, the slope temporarily levels out and then slopes down at 20, dropping an additional 38 m vertically before leveling out again. Find the skiers speed on the two level stretches.

A 10,000-kg Navy jet lands on an aircraft carrier and snags a cable to slow it down. The cable is attached to a spring with k = 40 kN/m. If the spring stretches 25 m to stop the plane, what was its landing speed?

A 120-g arrow is shot vertically from a bow whose effective spring constant is 430 N/m. If the bow is drawn 71 cm before shooting, to what height does the arrow rise?

In a railroad yard, a 35,000-kg boxcar moving at 7.5 m/s is stopped by a spring-loaded bumper mounted at the end of the level track. If k = 2.8 MN/m, how far does the spring compress in stopping the boxcar?

You work for a toy company, and youre designing a springlaunched model rocket. The launching apparatus has room for a spring that can be compressed 14 cm, and the rockets mass is 65 g. If the rocket is to reach an altitude of 35 m, what should you specify for the spring constant?

A 54-kg ice skater pushes off the wall of the rink, giving herself an initial speed of 3.2 m/s. She then coasts with no further effort. If the frictional coefficient between skates and ice is 0.023, how far does she go?

You push a 33-kg table across a 6.2-m-wide room. In the process, 1.5 kJ of mechanical energy gets converted to internal energy of the table/floor system. Whats the coefficient of kinetic friction between table and floor?

A particle slides along the frictionless track shown in Fig. 7.16, starting at rest from point A. Find (a) its speed at B, (b) its speed at C, and (c) the approximate location of its right-hand turning point. 0 1 2 3 4 5 6 7 8 9 10 11 12 Position, x (m) 3.8 m 2.6 m 1.3 m A B C Figure 7.16 Exercise 26

A particle slides back and forth on a frictionless track whose height as a function of horizontal position x is y = ax 2 , where a = 0.92 m-1 . If the particles maximum speed is 8.5 m/s, find its turning points.

A particle is trapped in a potential well described by U1x2 = 16x 2 - b, with U in joules, x in meters, and b = 4.0 J. Find the force on the particle when its at (a) x = 2.1 m, (b) x = 0, and (c) x = -1.4 m.

The reservoir at Northfield Mountain Pumped Storage Project is 214 m above the pump/generators and holds 2.1*1010 kg of water (see Application on p. 113). The generators can produce electrical energy at the rate of 1.08 GW. Find (a) the gravitational potential energy stored, taking zero potential energy at the generators, and (b) the length of time the station can generate power before the reservoir is drained.

The force in Fig. 7.14a is given by F S a = F0 nj, where F0 is a constant. The force in Fig. 7.14b is given by F S b = F01x/a2 nj, where the origin is at the lower left corner of the box, a is the width of the square box, and x increases horizontally to the right. Determine the work you would have to do to move an object around the perimeter of each each box, going clockwise at constant speed, starting at the lower left corner.

A 1.50-kg brick measures 20.0 cm * 8.00 cm * 5.50 cm. Taking the zero of potential energy when the brick lies on its broadest face, whats the potential energy (a) when the brick is standing on end and (b) when its balanced on its 8-cm edge? (Note: You can treat the brick as though all its mass is concentrated at its center.)

A carbon monoxide molecule can be modeled as a carbon atom and an oxygen atom connected by a spring. If a displacement of the carbon by 1.46 pm from its equilibrium position relative to the oxygen increases the molecules potential energy by 0.0125 eV, whats the spring constant?

A more accurate expression for the force law of the rope in Example 7.3 is F = -kx + bx 2 - cx 3 , where k and b have the values given in Example 7.3 and c = 3.1 N/m3 . Find the energy stored in stretching the rope 2.62 m. By what percentage does your result differ from that of Example 7.3?

For small stretches, the Achilles tendon can be modeled as an ideal spring. Experiments using a particular tendon showed that it stretched 2.66 mm when a 125-kg mass was hung from it. (a) Find the spring constant of this tendon. (b) How much would it have to stretch to store 50.0 J of energy?

The force exerted by an unusual spring when its compressed a distance x from equilibrium is F = -kx - cx 3 , where k = 220 N/m and c = 3.1 N/m3 . Find the stored energy when its been compressed 15 cm.

The force on a particle is given by F S = A ni/x 2 , where A is a positive constant. (a) Find the potential-energy difference between two points x1 and x2, where x1 7 x2. (b) Show that the potentialenergy difference remains finite even when x1 S .

A particle moves along the x-axis under the influence of a force F = ax 2 + b, where a and b are constants. Find the potential energy as a function of position, taking U = 0 at x = 0.

As a highway engineer, youre asked to design a runaway truck lane on a mountain road. The lane will head uphill at 30 and should be able to accommodate a 16,000-kg truck with failed brakes entering the lane at 110 km/h. How long should you make the lane? Neglect friction.

A spring of constant k, compressed a distance x, is used to launch a mass m up a frictionless slope at angle u. Find an expression for the maximum distance along the slope that the mass moves after leaving the spring.

A child is on a swing whose 3.2-m-long chains make a maximum angle of 50 with the vertical. Whats the childs maximum speed?

With x - x0 = h and a = g, Equation 2.11 gives the speed of an object thrown downward with initial speed v0 after its dropped a distance h: v = 2v2 0 + 2gh. Use conservation of mechanical energy to derive the same result.

The nuchal ligament is a cord-like structure that runs along the back of the neck and supports much of the heads weight in animals like horses and cows. The ligament is extremely stiff for small stretches, but loosens as it stretches further, thus functioning as a biological shock absorber. Figure 7.17 shows the forcedistance curve for a particular nuchal ligament; the curve can be modeled approximately by the expression F1x2 = 0.43x - 0.033x 2 + 0.00086x 3 , with F in kN and x in cm. Find the energy stored in the ligament when its been stretched (a) 7.5 cm and (b) 15 cm. 1.5 2.0 1.0 0.5 0.0 Force (kN) 0 2 4 8 10 12 14 6 Stretch (cm) Figure 7.17 Problem 42

A 200-g block slides back and forth on a frictionless surface between two springs, as shown in Fig. 7.18. The left-hand spring has k = 130 N/m and its maximum compression is 16 cm. The right-hand spring has k = 280 N/m. Find (a) the maximum compression of the right-hand spring and (b) the speed of the block as it moves between the springs. Figure 7.18 Problem 43

Automotive standards call for bumpers that sustain essentially no damage in a 4-km/h collision with a stationary object. As an automotive engineer, youd like to improve on that. Youve developed a spring-mounted bumper with effective spring constant 1.3 MN/m. The springs can compress up to 5.0 cm before damage occurs. For a 1400-kg car, what do you claim as the maximum collision speed?

A block slides on the frictionless loop-the-loop track shown in Fig. 7.19. Find the minimum height h at which it can start from rest and still make it around the loop. R h Figure 7.19 Problem 45

The maximum speed of the pendulum bob in a grandfather clock is 0.55 m/s. If the pendulum makes a maximum angle of 8.0 with the vertical, whats the pendulums length?

A mass m is dropped from height h above the top of a spring of constant k mounted vertically on the floor. Show that the springs maximum compression is given by 1mg/k211 + 11 + 2kh/mg2.

A particle with total energy 3.5 J is trapped in a potential well described by U = 7.0 - 8.0x + 1.7x 2 , where U is in joules and x in meters. Find its turning points.

(a) Derive an expression for the potential energy of an object subject to a force Fx = ax - bx 3 , where a = 5 N/m and b = 2 N/m3 , taking U = 0 at x = 0. (b) Graph the potentialenergy curve for x 7 0 and use it to find the turning points for an object whose total energy is -1 J.

In ionic solids such as NaCl (salt), the potential energy of a pair of ions takes the form U = b/rn - a/r, where r is the separation of the ions. For NaCl, a and b have the SI values 4.04 *10-28 and 5.52*10-98, respectively, and n = 8.22. Find the equilibrium separation in NaCl.

Repeat Exercise 19 for the case when the coefficient of kinetic friction on both slopes is 0.11, while the level stretches remain frictionless.

As an energy-efficiency consultant, youre asked to assess a pumped-storage facility. Its reservoir sits 140 m above its generating station and holds 8.5*109 kg of water. The power plant generates 330 MW of electric power while draining the reservoir over an 8.0-h period. Its efficiency is the percentage of the stored potential energy that gets converted to electricity. What efficiency do you report?

A spring of constant k = 340 N/m is used to launch a 1.5-kg block along a horizontal surface whose coefficient of sliding friction is 0.27. If the spring is compressed 18 cm, how far does the block slide?

A bug slides back and forth in a bowl 15 cm deep, starting from rest at the top, as shown in Fig. 7.20. The bowl is frictionless except for a 1.4-cm-wide sticky patch on its flat bottom, where the coefficient of friction is 0.89. How many times does the bug cross the sticky region? 1.4 cm 15 cm Figure 7.20 Problem 54

A 190-g block is launched by compressing a spring of constant k = 200 N/m by 15 cm. The spring is mounted horizontally, and the surface directly under it is frictionless. But beyond the equilibrium position of the spring end, the surface has frictional coefficient m = 0.27. . This frictional surface extends 85 cm, followed by a frictionless curved rise, as shown in Fig. 7.21. After its launched, where does the block finally come to rest? Measure from the left end of the frictional zone. Frictionless m = 0.27 Frictionless Figure 7.21 Problem 55

A block slides down a frictionless incline that terminates in a 45 ramp, as shown in Fig. 7.22. Find an expression for the horizontal range x shown in the figure as a function of the heights h1 and h2. 45 h1 h2 x Figure 7.22 Problem 56

An 840-kg roller-coaster car is launched from a giant spring with k = 31 kN/m into a frictionless loop-the-loop track of radius 6.2 m, as shown in Fig. 7.23. Whats the minimum spring compression that will ensure the car stays on the track? Figure 7.23 Problem 57

A particle slides back and forth in a frictionless bowl whose height is given by h = 0.18x 2 , with x and h in meters. Find the x coordinates of its turning points if the particles maximum speed is 47 cm/s

A child sleds down a frictionless hill whose vertical drop is 7.2 m. At the bottom is a level but rough stretch where the coefficient of kinetic friction is 0.51. How far does she slide across the level stretch?

A bug lands on top of the frictionless, spherical head of a bald man. It begins to slide down his head (Fig. 7.24). Show that the bug leaves the head when it has dropped a vertical distance onethird of the heads radius. Figure 7.24 Problem 60

A particle of mass m is subject to a force F S = 1a1x2 ni, where a is a constant. The particle is initially at rest at the origin and is given a slight nudge in the positive x-direction. Find an expression for its speed as a function of position x.

A block of weight 4.5 N is launched up a 30 inclined plane 2.0 m long by a spring with k = 2.0 kN/m and maximum compression 10 cm. The coefficient of kinetic friction is 0.50. Does the block reach the top of the incline? If so, how much kinetic energy does it have there? If not, how close to the top, along the incline, does it get?

Your engineering department is asked to evaluate the performance of a new 370-hp sports car. You know that 27% of the engines power can be converted to kinetic energy of the 1200-kg car, and that the power delivered is independent of the cars velocity. What do you report for the time it will take to accelerate from rest to 60 mi/h on a level road?

Your roommate is writing a science fiction novel and asks your advice about a plot point. Her characters are mining ore on the Moon and launching it toward Earth. Bins with 1500 kg of ore will be launched by a large spring, to be compressed 17 m. It takes a speed of 2.4 km/s to escape the Moons gravity. What do you tell her is an appropriate spring constant?

You have a summer job at your universitys zoology department, where youll be working with an animal behavior expert. Shes assigned you to study videos of different animals leaping into the air. Your task is to compare their power outputs as they jump. Youll have the mass m of each animal from data collected in the field. From the videos, youll be able to measure both the vertical distance d over which the animal accelerates when it pushes off the ground and the maximum height h it reaches. Your task is to find an algebraic expression for power in terms of these parameters.

Biomechanical engineers developing artificial limbs for prosthetic and robotic applications have developed a two-spring design for their replacement Achilles tendon. The first spring has constant k and the second ak, where a 7 1. When the artificial tendon is stretched from x = 0 to x = x1, only the first spring is engaged. For x 7 x1, a mechanism engages the second spring, giving a configuration like that described in part (a) of Chapter 4s Problem 62. Find an expression for the energy stored in the artificial tendon when its stretched a distance 2x1.

Blocks with different masses are pushed against a spring one at a time, compressing it different amounts. Each is then launched onto an essentially frictionless horizontal surface that then curves upward, still frictionless (like Fig. 7.21 but without the frictional part). The table below shows the masses, spring compressions, and maximum vertical height each block achieves. Determine a quantity that, when you plot h against it, should yield a straight line. Plot the data, determine a best-fit line, and use its slope to determine the spring constant. Mass m (g) 50.0 85.2 126 50.0 85.2 Compression x (cm) 2.40 3.17 5.40 4.29 1.83 Height h (cm) 10.3 11.2 19.8 35.2 3.81

Nuclear fusion is the process that powers the Sun. Fusion occurs when two low-mass atomic nuclei fuse together to make a larger nucleus, in the process releasing substantial energy. This is hard to achieve because atomic nuclei carry positive electric charge, and their electrical repulsion makes it difficult to get them close enough for the short-range nuclear force to bind them into a single nucleus. Figure 7.25 shows the potential-energy curve for fusion of two deuterons (heavy hydrogen nuclei). The energy is measured in million electron volts (MeV), a unit commonly used in nuclear physics, and the separation is in femtometers 11 fm = 10-15 m2. The force between the deuterons is zero at approximately a. 3 fm. b. 4 fm. c. 5 fm. d. the force is never zero.

Nuclear fusion is the process that powers the Sun. Fusion occurs when two low-mass atomic nuclei fuse together to make a larger nucleus, in the process releasing substantial energy. This is hard to achieve because atomic nuclei carry positive electric charge, and their electrical repulsion makes it difficult to get them close enough for the short-range nuclear force to bind them into a single nucleus. Figure 7.25 shows the potential-energy curve for fusion of two deuterons (heavy hydrogen nuclei). The energy is measured in million electron volts (MeV), a unit commonly used in nuclear physics, and the separation is in femtometers 11 fm = 10-15 m2. In order for initially two widely separated deuterons to get close enough to fuse, their kinetic energy must be about a. 0.1 MeV. b. 3 MeV. c. -3 MeV. d. 0.3 MeV.

Nuclear fusion is the process that powers the Sun. Fusion occurs when two low-mass atomic nuclei fuse together to make a larger nucleus, in the process releasing substantial energy. This is hard to achieve because atomic nuclei carry positive electric charge, and their electrical repulsion makes it difficult to get them close enough for the short-range nuclear force to bind them into a single nucleus. Figure 7.25 shows the potential-energy curve for fusion of two deuterons (heavy hydrogen nuclei). The energy is measured in million electron volts (MeV), a unit commonly used in nuclear physics, and the separation is in femtometers 11 fm = 10-15 m2. The energy available in fusion is the energy difference between that of widely separated deuterons and the bound deutrons after theyve fallen into the deep potential well shown in the figure. That energy is about a. 0.3 MeV. b. 1 MeV. c. 3.3 MeV. d. 3.6 MeV.

Nuclear fusion is the process that powers the Sun. Fusion occurs when two low-mass atomic nuclei fuse together to make a larger nucleus, in the process releasing substantial energy. This is hard to achieve because atomic nuclei carry positive electric charge, and their electrical repulsion makes it difficult to get them close enough for the short-range nuclear force to bind them into a single nucleus. Figure 7.25 shows the potential-energy curve for fusion of two deuterons (heavy hydrogen nuclei). The energy is measured in million electron volts (MeV), a unit commonly used in nuclear physics, and the separation is in femtometers 11 fm = 10-15 m2. When two deuterons are 4 fm apart, the force acting on them a. is repulsive. b. is attractive. c. is zero. d. cant be determined from the graph.