Chapter 6 and 7
A fan cart moves in the -x direction. The fan is on, and the force on the cart by the air is also in the -x direction. Is the work done by the air positive, negative, or zero?
Positive.
A tennis ball is moving in the -y direction. You hit it downward with a tennis racket. During the time your racket is in contact with the ball, do you do positive, negative, or zero work on the ball?
Positive.
If electricity cost 6.00/kWh (kilowatt-hour), how much would is cost you to run a 120 W stereo system 4.0 hours per day from 4.0 weeks?
$0.81
Find the net work done by friction on the body of a snake slithering in a complete circle of 3.93 m radius. The coefficient of friction between the ground and the snake is 0.25, and the snake's weight is 54.0 N.
-330J
You carry a 7 kg bag of groceries 1.2 m above the level floor at a constant velocity of 75 cm/s across a room that is 2.3 m room. How much work do you do on the bag in the process?
0.0J
A stock person at the local grocery store has a job consisting of the following five segments: (1) picking up boxes of tomatoes from the stockroom floor (2) accelerating to a comfortable speed (3) carrying the boxes to the tomato display at constant speed (4) decelerating to a stop (5) lowering the boxes slowly to the floor. During which of the five segments of the job does the stock person do positive work on the boxes?
1 and 2.
A potential energy function is given by U(x) = (3.00 N/m)x - (1.00 N/m^3)x^3. At what position or positions is the force equal to zero?
1.00 m and -1.00 m
A tennis balls bounces on the floor three times. If each time it loses 22.0% of its energy due to heating, how high does it rise after the third bounce, provided we released it 2.3 m from the floor?
110 cm
A block slides down a frictionless inclined ramp. If the ramp angle is 17.0 degrees and its length is 30.0 m, find the speed of the block as it reaches the bottom of the ramp, assuming it started sliding from rest at the top.
13.1 m/s
A car on a roller coaster starts at zero speed at an elevation above the ground of 26 m. It coasts down a slope, and then climbs a hill. The top of the hill is at an elevation of 16 m. What is the speed of the car at the top of the hill? Neglect any frictional effects
14 m/s
A child pulls on a wagon with a horizontal force of 75N. If the wagon moves horizontally a total of 42m in 3.0 min, what is the average power generated by the child?
18W
How long will it take a 7.08 hp motor to lift 250 kg beam directly upward at constant velocity from the ground to a height of 45.0 m? Assume frictional forces are negligible. (1 hp = 746 W)
20.9 s
Consider the motion is a 1.00-kg particle that moves with potential energy given by U(x) = (-2.00 J * m)/x + (4.00 J * m^2)/x^2. Suppose the particle is moving with a speed of 3.00 m/s when it is located at x = 1.00 m. What is the speed of the object when it is located at x = 5.00 m?
3.67 m/s
On flat ground, a 70 kg person requires about 300 W of metabolic power to walk at a steady pace of 5.0 km/h (1.4 m/s). Using the same metabolic power output, that person can bicycle over the same ground at 15 km/h. A 70 kg person walks at a steady pace of 5.0 km/h on a treadmill at a 5.0% grade. (That is, the vertical distance covered is 5.0% of the horizontal distance covered.) If we assume the metabolic power required is equal to that required for walking on a flat surface plus the rate of doing work for the vertical climb, how much power is required?
350 W
A worker lifts a 20.0-kg bucket of concrete from the ground up to the top of a 20.0-m tall building. The bucket is initially at rest, but is traveling at 4.0 m/s when it reaches the top of the building. What is the minimum amount of work that the worker did in lifting the bucket?
4.08 kJ
A 1000.0 kg car is moving at 15 km/h. If a 2000.0 kg truck has 18 times the kinetic energy of the car, how fast is the truck moving?
45 km/h
A force F = 12Ni - 10Nj acts on an object. How much work does this force do as the object moves from the origin to the point r = 13mi + 11mj
46J
A spring-loaded dart gun is used to shoot a dart straight up into the air, and the dart reaches a maximum height of 24 meters. The same dart is shot up a second time from the same gun, but this time the spring is compressed only half as far (compared to the first shot). How far up does the dart go this time (neglect all friction and assume the spring is ideal)?
60. m
A person pushes horizontally on a heavy box and slides it across the level floor at constant velocity. The person pushes with a 60.0N force for the first 6.88m, at which time he begins to tire. The force he exerts then starts to decrease linearly from 60.0N to 0.00N across the remaining 6.88m. How much total work did the person do on the box?
619J
A traveler pulls on a suitcase strap at an angle 36 degrees above the horizontal. If 908 J of work are done by the strap while moving the suitcase a horizontal distance of 15 m, what is the tension in the strap?
75N
The restoring force of three springs is measured as they are stretched. Which spring has the largest spring constant?
A.
A ball travels straight up, coming to a stop after it has risen a distance h. Which equation (/\Esys = W) applies to the system of the ball alone?
A. -1/2mv^2i = -mgh
A thrown ball heads straight up. SYSTEM: Ball + Earth What is the work done by the surroundings?
A. 0
The graph shows the potential energy for a particle that moves along the x-axis. The only force that acts on the particle is the force associated with U. The particle is initially at x = d and moves in the negative x-direction. At which of the labeled x-coordinates is the particle slowing down?
A. x = a
Two stones, one of mass m and the other of mass 2m, are thrown directly upward with the same velocity at the same time from ground level and feel no air resistance. Which statement about these stones is true?
At its highest point, the heavier stone will have twice as much gravitational potential energy as the lighter one because it is twice as heavy.
A ball travels straight up, coming to a stop after it has risen a distance h. Which equation (/\Esys = W) applies to the system of the ball + earth?
B. -1/2mv^2i + mgh = 0
An elevator is being lifted at a constant speed by a steel cable attached to an electronic motor. Which statement is correct?
B. The cable does positive work on the elevator, and the elevator does negative work on the cable.
The graph shows the potential energy for a particle that moves along the x-axis. The only force that acts on the particle is the force associated with U. The particle is initially a x = d and moves in the negative x-direction. At which of the labeled x-coordinates does the particle have the greatest speed?
B. x = b
Two identical balls are thrown directly upward, ball A at speed v and ball B at speed 2v, and they feel no air resistance. Which statement about these balls is correct?
Ball B will go four times as high ball A because it has four times the initial kinetic energy.
A thrown ball heads straight up. SYSTEM: Ball What is the work done by the surroundings?
C. -mg/\y
A thrown ball heads straight up. What is the work done by the surroundings for the system of the ball only?
C. -mg/\y
A thrown ball heads straight up. SYSTEM: Ball How did the kinetic energy of the system change?
C. /\K < 0
A thrown ball heads straight up. SYSTEM: Ball + Earth How did the kinetic energy of the system change?
C. /\K < 0
A thrown ball heads straight up. How did the kinetic energy of the system change?
C. /\K < 0
The two ramps shown are both frictionless. The heights y1 and y2 are the same for each ramp. A block of mass m is released from rest at the left-hand and of each ramp. Which block arrives at the right-hand end with the greater speed?
C. Both blocks arrive at the right-hand end with the same speed.
Consider a block falling onto a vertical spring. States: A: Block 0.8 m above floor. B: Block just touching top of spring. C: Block 0.3 m above floor. To find the maximum compression of the spring. What should we pick for initial and final states?
C. Initial A, final C
An elevator is being lowered at a constant speed by a steel cable attached to an electric motor. Which statement is correct?
C. The cable does negative work on the elevator, and the elevator does positive work on the cable.
A tractor driving at a constant speed pulls a sled loaded with firewood. There is friction between the sled and the road. The total work done on the sled after it has moved a distance d is:
C. Zero
A block is released from rest on a frictionless incline as shown. When the moving block is in contact with the spring and compressing it, what is happening to the gravitational potential energy Ugrow and the elastic potential energy Uel?
D. Ugrav is decreasing: Uel is increasing.
The graph shows the potential energy for a particle that moves along the x-axis. The only force that acts on the particle is the force associated with U. The particle is initially x = d and moves in the negative x-direction. At which of the labeled x-coordinates is there zero force on the particle?
D. x = b and x = d
Two iceboats (one of mass m, one of mass 2m) hold a race on a frictionless, horizontal, frozen lake. Both iceboats start at rest, and the wind exerts the same constant force on both iceboats. Which iceboat crosses the finish line in more kinetic energy (K)?
E. They both cross the finish line with the same kinetic energy.
As part of your daily workout, you lie on your back and push with your feet against a platform attached to two stiff ideal springs arranged side by side so that they are parallel to each other. When you push the platform, you compress the springs. You do 85.0 J of work when you compress the springs 0.200 m from their uncompressed length. What magnitude of force must you apply to hold the platform in this position? Express your answer with the appropriate units. How much additional work must you do to move the platform 0.200 m farther? Express your answer with the appropriate units. What maximum force must you apply to move the platform to the position in Part B? Express your answer with the appropriate units.
F = 850 N W = 255 J F max = 1700 N
A factory worker pushes a 30.0 kg crate a distance of 4.3 m along a level floor at constant velocity by pushing downward at an angle of 30∘ below the horizontal. The coefficient of kinetic friction between the crate and floor is 0.25. What magnitude of force must the worker apply to move the crate at constant velocity? Express your answer in newtons. How much work is done on the crate by this force when the crate is pushed a distance of 4.3 m? Express your answer in joules. How much work is done on the crate by friction during this displacement? Express your answer in joules. How much work is done by the normal force? Express your answer in joules. How much work is done by gravity? Express your answer in joules. What is the total work done on the crate? Express your answer in joules.
F = 99 N W = 370 J Wf = -370 J Wnf = 0 J Wg = 0 J Wnet = 0 J
The potential energy of two atoms in a diatomic molecule is approximated by U(r)=a/r12−b/r6, where r is the spacing between atoms and a and b are positive constants. Find the component of force along the line connecting the two atoms, Fr(r), on one atom as a function of r. Express your answer in terms of the variables a, b, and r. Find the equilibrium distance between the two atoms. Express your answer in terms of the variables a and b. Is this equilibrium stable? Suppose the distance between the two atoms is equal to the equilibrium distance found in part A. 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. Express your answer in terms of the variables a and b. For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is 1.13×10−10m and the dissociation energy is 1.54×10−18J per molecule. Find the value of the constant a. Express your answer in joules times meter in the twelth power. Find the value of the constant b. Express your answer in joules times meter in the sixth power.
Fr(r) = 12a/r^13 - 6b/r^7 rmin = 6th root (2a/b) yes E0 = b^2/4a a = 6.68 * 10^-138 J * m^12 b = 6.41 * 10^-78 J*m^6
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? Express your answer in terms of C6 and x. Is this force attractive or repulsive?
Fx = -6C6/x^7 attractive
You slam on the brakes of your car in a panic, and skid a certain distance on a straight, level road. If you had been traveling twice as fast, what distance would the car have skidded, under the same conditions?
It would have skidded 4 times farther.
A mass is pressed against (but is not attached to) an ideal horizontal spring on a frictionless horizontal surface. After being released from rest, the mass acquires a maximum speed v and a maximum kinetic energy K. If instead the mass initially compresses the spring twice as far,
Its maximum speed will be v * sqrt(2) and its maximum kinetic energy will be 2K.
A ball drops some distance and loses 30 J of gravitational potential energy. Do not ignore air resistance. How much kinetic energy did the ball gain?
Less than 30 J
A figure skater slides in the -x direction along the ice, toward her partner. When she gets close he pushes on her in the +x direction, to slow her down. Does he do positive, negative, or zero work?
Negative.
A skydiver falls toward the Earth, with his parachute open. During his fall, does the force by the air does positive, negative, or zero work on the system of skydiver plus parachute?
Negative.
A 20.0 kg rock is sliding on a rough, horizontal surface at 8.00 m/s and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is 0.200. What average thermal power is produced as the rock stops? Express your answer with the appropriate units.
P = 157 W
When its 70 kW engine is generating full power, a small single-engine airplane with mass 700 kg gains altitude at a rate of 2.5 m/s. What fraction of the engine power is being used to make the airplane climb? (The remainder is used to overcome the effects of air resistance and of inefficiencies in the propeller and engine.) Express your answer as a percentage to two significant figures.
Puseful /Pfull = 25%
You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of 465 N in the rope. If Sneezy hangs from a similar rope while delivering presents at the earth's equator, what will the tension in it be? (Recall that the earth is rotating about an axis through its north and south poles.) Express your answer with the appropriate units.
T = 463 N
A potential energy function for system 1 is given by U1 (x) = Cx^2 + Bx^3. The potential energy function for system 2 is given by U2 (x) = A + Cx^2 + Bx^3, where A is a positive quantity. How does the force on system 1 relate to the force on system 2 at a given position?
The force is identical on the two systems.
Consider a plot of the displacement (x) as a function of the applied force (F) for an ideal elastic spring. The slope of the curve would be:
The reciprocal of the spring constant.
A constant horizontal pull acts on a sled on a horizontal frictionless ice pond. The sled starts from rest. When the pull acts over a distance x, the sled acquires a speed v and a kinetic energy K. If the same pull instead acts over twice this distance,
The sled's speed will be v * sqrt(2) and its kinetic energy will be 2K.
In a physics lab experiment, one end of a horizontal spring that obeys Hooke's law is attached to a wall. The spring is compressed x0 = 0.400 m, and a block with mass 0.300 kg is attached to it. The spring is then released, and the block moves along a horizontal surface. Electronic sensors measure the speed v of the block after it has traveled a distance d from its initial position against the compressed spring. The measured values are listed in the table below. The data show that the speed v of the block increases and then decreases as the spring returns to its unstretched length. Explain why this happens, in terms of the work done on the block by the forces that act on it. Use the work-energy theorem to derive an expression for v2 in terms of d. Do not substitute the value of x0 into the expression. Express your answer in terms of some or all of the variables k, m, d, x0, μk, and the acceleration due to gravity g. Select the correct graph of the data as v2v2 (vertical axis) versus dd (horizontal axis). Find the block's maximum speed v using the equation from the graph in part C. Express your answer to three significant figures and include the appropriate units. Find the value of d at which the maximum speed v occurs using the equation from the graph in part C. Express your answer to three significant figures and include the appropriate units. By comparing the equation from the graph to the formula you derived in part B, calculate the force constant k for the spring. Express your answer to three significant figures and include the appropriate units. By comparing the equation from the graph to the formula you derived in part B, calculate the coefficient of kinetic friction for the friction force that the surface exerts on the block. Express your answer using three significant figures.
The spring force is initially greater than friction, so the block accelerates forward. But eventually the spring force decreases enough so that it is less than the force of friction, and the block then slows down (decelerates). v^2(t) = -k/m * d^2 + 2d(k/mXo - mg) v = 1.29 m/s d = 0.204 m k = 12.0 N/m mk = 0.799
T or F: If a force always acts perpendicular to an object's direction of motion, that force cannot change the object's kinetic energy.
True
T or F: When an object is solely under the influence of conservative forces, the sum of its kinetic and potential energies does not change.
True
A 27.0 kg child plays on a swing having support ropes that are 2.10 m long. A friend pulls her back until the ropes are 45.0 ∘ from the vertical and releases her from rest. What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing? Express your answer in joules. How fast will she be moving at the bottom of the swing? Express your answer in meters per second. How much work does the tension in the ropes do as the child swings from the initial position to the bottom? Express your answer in joules.
U = 163 J v = 3.47 m/s W = 0 J
A force of 570 N keeps a certain ideal spring stretched a distance of 0.300 m. What is the potential energy of the spring when it is stretched 0.300 m? Express your answer with the appropriate units. What is its potential energy when it is compressed 9.00 cm? Express your answer with the appropriate units.
U1 = 86 J U2 = 7.7 J
Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.5×106 N, one an angle 18 ∘ west of north and the other an angle 18 ∘ east of north, as they pull the tanker a distance 0.64 km toward the north. What is the total work they do on the supertanker? Express your answer in joules.
W = 1.8 * 10^9 J
A 6.90 kg watermelon is dropped from rest from the roof of a 19.0 m-tall building and feels no appreciable air resistance. Calculate the work done by gravity on the watermelon during its displacement from the roof to the ground. Express your answer with the appropriate units. Just before it strikes the ground, what is the watermelon's kinetic energy? Express your answer with the appropriate units. Just before it strikes the ground, what is the watermelon's speed? Express your answer with the appropriate units. Would the answer in part A be different if there were appreciable air resistance? Would the answer in part B be different if there were appreciable air resistance? Would the answer in part C be different if there were appreciable air resistance?
W = 1280 J K = 1280 J v = 19.3 m/s no yes yes
A 0.60 kg book slides on a horizontal table. The kinetic friction force on the book has magnitude 1.6 N. How much work is done on the book by friction during a displacement of 3.0 m to the left? Express your answer with the appropriate units. 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? Express your answer with the appropriate units. What is the total work done on the book by friction during the complete round trip? Express your answer with the appropriate units. On the basis of your answer to part C, would you say that the friction force is conservative or nonconservative?
W1 = -4.8 J W2 = -4.8 J W3 = -9.6 J noncenservative
A luggage handler pulls a 20.0 kg suitcase up a ramp inclined at 33.0 ∘ above the horizontal by a force F⃗ of magnitude 169 N that acts parallel to the ramp. The coefficient of kinetic friction between the ramp and the incline is μk = 0.300. The suitcase travels 3.70 m along the ramp. Calculate the work done on the suitcase by F⃗ . Express your answer with the appropriate units. Calculate the work done on the suitcase by the gravitational force. Express your answer with the appropriate units. Calculate the work done on the suitcase by the normal force. Express your answer with the appropriate units.
WF = 625 J Ww = -395 J Wn = 0 J Wf = -182 J v = 2.19 m/s
Is it possible for a system to have negative potential energy?
Yes, since the choice of the zero of potential energy is arbitrary.
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)=(5.50J/m2)x2−(3.60J/m3)y3. What is the magnitude of the acceleration of the block when it is at the point x = 0.23 m, y = 0.70 m? Express your answer with the appropriate units. What is the direction of the acceleration of the block when it is at the point x = 0.23 m, y = 0.70 m? Express your answer in degrees.
a = 147 m/s^2 theta = 116 degrees counterclockwise from the +x-axis
You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1300 kg car moving at 0.64 m/s is to compress the spring no more than 0.079 m before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass. Express your answer in newtons per meter.
k = 8.5 * 10^4 N/m
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 (Figure 1). When released, the block moves on a horizontal tabletop for 1.00 m before coming to rest. The spring constant k is 100 N/m. What is the coefficient of kinetic friction μk between the block and the tabletop?
mk = 0.41
On an essentially frictionless, horizontal ice rink, a skater moving at 5.0 m/s encounters a rough patch that reduces her speed by 50 % due to a friction force that is 30 % of her weight. Use the work-energy theorem to find the length of this rough patch. Express your answer in meters.
s = 3.2 m
You are an industrial engineer with a shipping company. As part of the package-handling system, a small box with mass 1.30 kg is placed against a light spring that is compressed 0.280 m. The spring, whose other end is attached to a wall, has force constant k = 48.0 N/m. The spring and box are released from rest, and the box travels along a horizontal surface for which the coefficient of kinetic friction with the box is μk = 0.300. When the box has traveled 0.280 m and the spring has reached its equilibrium length, the box loses contact with the spring. What is the speed of the box at the instant when it leaves the spring? Express your answer with the appropriate units. What is the maximum speed of the box during its motion? Express your answer with the appropriate units.
v = 1.12 m/s vmax = 1.22 m/s
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 45 ∘ with the vertical. Air resistance is negligible. What is the speed of the rock when the string passes through the vertical position? Express your answer in meters per second. What is the tension in the string when it makes an angle of 45∘ with the vertical? Express your answer in newtons. What is the tension in the string as it passes through the vertical? Express your answer in newtons.
v = 2.1 m/s T45 = 0.83 N Tvert = 1.9 N
A 0.230 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. What is the speed of the potato at the lowest point of its motion? Express your answer in meters per second. What is the tension in the string at this point? Express your answer in newtons.
v = 7.00 m/s T = 6.76 N
A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of 50.0 ∘ above the horizontal. The glider has mass 9.00×10−2 kg. The spring has 600 N/m and negligible mass. When the spring is released, the glider travels a maximum distance of 1.50 m along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring. What distance was the spring originally compressed? Express your answer in meters. When the glider has traveled along the air track 0.500 m from its initial position against the compressed spring, is it still in contact with the spring? What is the kinetic energy of the glider at this point? Express your answer in joules.
x = 5.8 * 10^-2 m no K = 0.675 J