47. The force acting on a particle varies as in Figure P5.47.
Find the work done by the force as the particle moves (a) fromx0 tox8.00 m, (b) fromx8.00 m tox 10.0 m, and (c) fromx0 tox10.0 m.
2 4 6 8 10 x(m)
–2 –4 2 4 6 Fx(N)
Figure P5.47
51.An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from zero to 230 N. (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do in pulling the bow?
52.A block of mass 12.0 kg slides from rest down a friction- less 35.0° incline and is stopped by a strong spring with k3.00104N/m. The block slides 3.00 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed?
53.(a) A 75-kg man steps out a window and falls (from rest) 1.0 m to a sidewalk. What is his speed just before his feet strike the pavement? (b) If the man falls with his knees and ankles locked, the only cushion for his fall is an ap- proximately 0.50-cm give in the pads of his feet. Calculate the average force exerted on him by the ground in this sit- uation. This average force is sufficient to cause damage to cartilage in the joints or to break bones.
54.A toy gun uses a spring to project a 5.3-g soft rubber sphere horizontally. The spring constant is 8.0 N/m, the barrel of the gun is 15 cm long, and a constant frictional force of 0.032 N exists between barrel and projectile. With what speed does the projectile leave the barrel if the spring was compressed 5.0 cm for this launch?
55.Two objects are connected by a light string passing over a light, frictionless pulley as in Figure P5.55. The 5.00-kg object is released from rest at a point 4.00 m above the floor. (a) Determine the speed of each object when the two pass each other. (b) Determine the speed of each ob- ject at the moment the 5.00-kg object hits the floor.
(c) How much higher does the 3.00-kg object travel after the 5.00-kg object hits the floor?
0 2 4 6 8 10 12 14 16
1 2 3 Fx(N)
x(m)
Figure P5.48
56.Two blocks,AandB(with mass 50 kg and 100 kg, respec- tively), are connected by a string, as shown in Figure P5.56.
h 4.00 m m2 3.00 kg
m1 5.00 kg
Figure P5.55
50 kg 100 kg
37°
A
B
Figure P5.56 48.An object is subject to a force Fxthat varies with position
as in Figure P5.48. Find the work done by the force on the object as it moves (a) from x0 to x5.00 m, (b) from x5.00 m to x10.0 m, and (c) from x10.0 m to x15.0 m. (d) What is the total work done by the force over the distance x0 to x15.0 m?
49. The force acting on an object is given by Fx(8x16) N, where xis in meters. (a) Make a plot of this force versus x from x0 to x3.00 m. (b) From your graph, find the net work done by the force as the object moves from x0 to x3.00 m.
ADDITIONAL PROBLEMS
50. A 2.0-m-long pendulum is released from rest when the support string is at an angle of 25° with the vertical. What is the speed of the bob at the bottom of the swing?
Problems 121
The pulley is frictionless and of negligible mass. The coef- ficient of kinetic friction between blockAand the incline ismk0.25. Determine the change in the kinetic energy of blockAas it moves fromto, a distance of 20 m up the incline if the system starts from rest.
57.A 200-g particle is released from rest at point Aon the in- side of a smooth hemispherical bowl of radius R30.0 cm (Fig. P5.57). Calculate (a) its gravitational potential en- ergy at Arelative to B, (b) its kinetic energy at B, (c) its speed at B, (d) its potential energy at Crelative to B, and (e) its kinetic energy at C.
with the vertical. (See Fig. P5.61.) In the figure, D 50.0 m, F110 N, L40.0 m, andu50.0°. (a) With what minimum speed must Jane begin her swing in order to just make it to the other side? (Hint:First determine the potential energy that can be associated with the wind force.
Because the wind force is constant, use an analogy with the constant gravitational force.) (b) Once the rescue is com- plete, Tarzan and Jane must swing back across the river.
With what minimum speed must they begin their swing?
58.Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is 1 kilocalorie, which we define in Chapter 11 as 1 kcal4 186 J. Metabolizing 1 gram of fat can release 9.00 kcal. A student decides to try to lose weight by exercising. She plans to run up and down the stairs in a football stadium as fast as she can and as many times as necessary. Is this in itself a practical way to lose weight? To evaluate the program, suppose she runs up a flight of 80 steps, each 0.150 m high, in 65.0 s. For sim- plicity, ignore the energy she uses in coming down (which is small). Assume that a typical efficiency for human mus- cles is 20.0%. This means that when your body converts 100 J from metabolizing fat, 20 J goes into doing mechani- cal work (here, climbing stairs). The remainder goes into internal energy. Assume the student’s mass is 50.0 kg.
(a) How many times must she run the flight of stairs to lose 1 pound of fat? (b) What is her average power output, in watts and in horsepower, as she is running up the stairs?
59. In terms of saving energy, bicycling and walking are far more efficient means of transportation than is travel by automobile. For example, when riding at 10.0 mi/h, a cyclist uses food energy at a rate of about 400 kcal/h above what he would use if he were merely sitting still.
(In exercise physiology, power is often measured in kcal/h rather than in watts. Here, 1 kcal1 nutritionist’s Calorie4 186 J.) Walking at 3.00 mi/h requires about 220 kcal/h. It is interesting to compare these values with the energy consumption required for travel by car. Gaso- line yields about 1.30108J/gal. Find the fuel economy in equivalent miles per gallon for a person (a) walking and (b) bicycling.
60.An 80.0-N box is pulled 20.0 m up a 30° incline by an ap- plied force of 100 N that points upwards, parallel to the incline. If the coefficient of kinetic friction between box and incline is 0.220, calculate the change in the kinetic energy of the box.
61.Jane, whose mass is 50.0 kg, needs to swing across a river filled with crocodiles in order to rescue Tarzan, whose mass is 80.0 kg. However, she must swing into aconstanthorizon- tal wind force:Fon a vine that is initially at an angle ofu
2R/3 C
B R A
Figure P5.57
L Jane
D θ φ Wind TarzanF
Figure P5.61
62.A hummingbird is able to hover because, as the wings move downwards, they exert a downward force on the air. Newton’s third law tells us that the air exerts an equal and opposite force (upwards) on the wings. The average of this force must be equal to the weight of the bird when it hovers. If the wings move through a dis- tance of 3.5 cm with each stroke, and the wings beat 80 times per second, determine the work performed by the wings on the air in 1 minute if the mass of the hum- mingbird is 3.0 grams.
63.A child’s pogo stick (Fig. P5.63) stores energy in a spring (k2.50104N/m). At position(x1 0.100 m), the spring compression is a maximum and the child is momentarily at rest. At position(x0), the spring is
x1
x2
Figure P5.63
122 Chapter 5 Energy
relaxed and the child is moving upwards. At position, the child is again momentarily at rest at the top of the jump. Assuming that the combined mass of child and pogo stick is 25.0 kg, (a) calculate the total energy of the system if both potential energies are zero at x0, (b) determinex2, (c) calculate the speed of the child at x0, (d) determine the value ofxfor which the kinetic energy of the system is a maximum, and (e) obtain the child’s maximum upward speed.
64. A 2.00-kg block situated on a rough incline is connected to a spring of negligible mass having a spring constant of 100 N/m (Fig. P5.64). The block is released from rest when the spring is unstretched, and the pulley is friction- less. The block moves 20.0 cm down the incline before coming to rest. Find the coefficient of kinetic friction be- tween block and incline.
piece of uniform elastic cord tied to a harness around his body to stop his fall at a point 10.0 m above the ground.
Model his body as a particle and the cord as having negli- gible mass and a tension force described by Hooke’s force law. In a preliminary test, hanging at rest from a 5.00-m length of the cord, the jumper finds that his body weight stretches it by 1.50 m. He will drop from rest at the point where the top end of a longer section of the cord is at- tached to the stationary balloon. (a) What length of cord should he use? (b) What maximum acceleration will he experience?
67. The system shown in Figure P5.67 consists of a light, inex- tensible cord, light frictionless pulleys, and blocks of equal mass. Initially, the blocks are at rest the same height above the ground. The blocks are then released. Find the speed of block A at the moment when the vertical separa- tion of the blocks is h.
65. A loaded ore car has a mass of 950 kg and rolls on rails with negligible friction. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at 30.0° above the horizontal. The car accelerates uniformly to a speed of 2.20 m/s in 12.0 s and then con- tinues at constant speed. (a) What power must the winch motor provide when the car is moving at constant speed?
(b) What maximum power must the motor provide?
(c) What total energy transfers out of the motor by work by the time the car moves off the end of the track, which is of length 1 250 m?
66. A daredevil wishes to bungee-jump from a hot-air balloon 65.0 m above a carnival midway (Fig. P5.66). He will use a
68.A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from four identical spiral springs under tension, one near each corner of the shelf. Each tray has a mass of 580 g and is rectangular, 45.3 cm by 35.6 cm, and 0.450 cm thick. (a) Show that the top tray in the stack can always be at the same height above the floor, however many trays are in the dispenser. (b) Find the spring con- stant each spring should have in order for the dispenser to function in this convenient way. Is any piece of data unnecessary for this determination?
69.In bicycling for aerobic exercise, a woman wants her heart rate to be between 136 and 166 beats per minute. Assume that her heart rate is directly proportional to her mechan- ical power output. Ignore all forces on the woman-plus- bicycle system, except for static friction forward on the drive wheel of the bicycle and an air resistance force pro- portional to the square of the bicycler’s speed. When her speed is 22.0 km/h, her heart rate is 90.0 beats per minute. In what range should her speed be so that her heart rate will be in the range she wants?
70.In a needle biopsy, a narrow strip of tissue is extracted from a patient with a hollow needle. Rather than being pushed by hand, to ensure a clean cut the needle can be fired into the patient’s body by a spring. Assume the nee- dle has mass 5.60 g, the light spring has force constant 375 N/m, and the spring is originally compressed 8.10 cm to project the needle horizontally without friction. The tip of the needle then moves through 2.40 cm of skin and soft tissue, which exerts a resistive force of 7.60 N on it.
Next, the needle cuts 3.50 cm into an organ, which exerts a backward force of 9.20 N on it. Find (a) the maximum speed of the needle and (b) the speed at which a flange 37.0°
2.00 kg k = 100 N/m
Figure P5.64
Figure P5.66 Bungee jumping.
© Jamie Budge/Corbis
A B
Figure P5.67
Problems 123
on the back end of the needle runs into a stop, set to limit the penetration to 5.90 cm.
71. The power of sunlight reaching each square meter of the Earth’s surface on a clear day in the tropics is close to 1 000 W. On a winter day in Manitoba, the power concen- tration of sunlight can be 100 W/m2. Many human ac- tivities are described by a power-per-footprint-area on the order of 102W/m2 or less. (a) Consider, for example, a family of four paying $80 to the electric company every 30 days for 600 kWh of energy carried by electric transmission
to their house, with floor area 13.0 m by 9.50 m. Compute the power-per-area measure of this energy use. (b) Con- sider a car 2.10 m wide and 4.90 m long traveling at 55.0 mi/h using gasoline having a “heat of combustion” of 44.0 MJ/kg with fuel economy 25.0 mi/gallon. One gallon of gasoline has a mass of 2.54 kg. Find the power-per-area measure of the car’s energy use. It can be similar to that of a steel mill where rocks are melted in blast furnaces.
(c) Explain why the direct use of solar energy is not practi- cal for a conventional automobile.
124
O U T L I N E
6.1 Momentum and Impulse 6.2 Conservation of
Momentum 6.3 Collisions
6.4 Glancing Collisions 6.5 Rocket Propulsion
Momentum and Collisions
What happens when two automobiles collide? How does the impact affect the motion of each vehicle, and what basic physical principles determine the likelihood of serious injury?
Why do we have to brace ourselves when firing small projectiles at high velocity?
To begin answering such questions, we introducemomentum. Intuitively, anything that has a lot of momentum is going to be hard to stop. Physically, the more momentum an object has, the more force has to be applied to stop it in a given time. This concept leads to one of the most powerful principles in physics:conservation of momentum. Using this law, complex collision problems can be solved without knowing much about the forces involved during con- tact. We’ll also be able to derive information about the average force delivered in an impact.
6.1 MOMENTUM AND IMPULSE
In physics, momentum has a precise definition. A slowly moving brontosaurus has a lot of momentum, but so does a little hot lead shot from the muzzle of a gun. We therefore expect that momentum will depend on an object’s mass and velocity.
The linear momentum of an object of mass mmoving with velocity is the product of its mass and velocity :
[6.1]
SI unit: kilogram-meter per second (kgm/s)
Doubling either the mass or the velocity of an object doubles its momentum; dou- bling both quantities quadruples its momentum. Momentum is a vector quantity with the same direction as the object’s velocity. Its components are given in two dimensions by
pxmvx pymvy
where px is the momentum of the object in the x- direction and pyits momentum in the y-direction.
Changing the momentum of an object requires the application of a force. This is, in fact, how Newton originally stated his second law of motion. Starting from the more common version of the second law, we have
[6.2]
where the mass mand the forces are assumed constant. The quantity in parenthe- ses is just the momentum, so we have the following result:
The change in an object’s momentum divided by the elapsed time t equals the constant net force acting on the object:
[6.3]
:p
t change in momentum time interval :Fnet
:F
net
:p F:netma:m :v
t (m:v) t
:p m:v
:v p:
Linear momentum