Download Chapter 5: Forces and Motion II

Questions and Problems 189
Questions and Problems
In a few problems, you are given more data than you actually need;
in a few other problems, you are required to supply data from
your general knowledge, outside sources, or informed estimate.
Interpret as significant all digits in numerical values that
have trailing zeros and no decimal points. For all problems, use
g = 9.80 m>s2 for the free-fall acceleration due to gravity.
• Basic, single-concept problem
•• Intermediate-level problem, may require synthesis of concepts and multiple steps
••• Challenging problem
SSM Solution is in Student Solutions Manual
Conceptual Questions
1. •Complete the sentence: The static frictional force between
two surfaces is (never/sometimes/always) less than the normal
force. Explain your answer.
2. •You want to push a heavy box of books across a rough
floor. You know that the maximum value of the coefficient
of static friction (ms) is larger than the maximum value of the
­coefficient of kinetic friction (mk). Should you push the box for
a short distance, rest, push the box another short distance, and
then repeat the process until the box is where you want it, or
will it be easier to keep pushing the box across the floor once
you get it moving?
3. •If the force of friction always opposes the sliding of an
­object, how then can a frictional force cause an object to increase in speed? SSM
4. •A solid rectangular block has sides of three different areas.
You can choose to rest any of the sides on the floor as you
­apply a horizontal force to the block. Does the choice of side
on the floor affect how hard it is to push the block? Explain
your answer.
5. •You’re trying to press a book against a spot on the wall with
your hand. As you get tired, you exert less force, but the book
remains in the same spot on the wall. Do each of the following forces increase, decrease, or not change in magnitude when
you reduce the force you are applying to the book: (a) weight,
(b) normal force, (c) frictional force, and (d) maximum static
frictional force?
6. •For an object moving in a circle, which of the following quantities are zero over one revolution: (a) displacement,
(b) average velocity, (c) average acceleration, (d) instantaneous
velocity, and (e) instantaneous centripetal acceleration?
for different people, in spite of the fact that their joints have the
same coefficient of friction? For simplicity, assume the surfaces
in the knee are flat and horizontal.
11. •As a skydiver falls faster and faster through the air, does
his acceleration increase, decrease, or remain the same? Explain
your answer. SSM
12. •Why do raindrops fall from the sky at different speeds?
Explain your answer.
13. •Why might your car start to skid if you drive too fast
around a curve?
14. •What distinguishes the forces that act on a car driving over
the top of a hill from those acting on a car driving through a
dip in the road? Explain how the forces relate to the sensations
passengers in the car experience during each situation.
15. •Explain how you might measure the centripetal acceleration of a car rounding a curve. SSM
16. •An object of mass M1 rests on a horizontal table; friction
exists between the object and the table (Figure 5-21). A ring of
mass MR is tied by massless strings both to the wall and to the
object as shown. A second object of mass M2 hangs from the
ring. Draw free-body diagrams for the situation.
Friction
M1
MR
M2
Figure 5-21
Problem 16
17. •Explain why curves in roads and cycling velodromes are
banked. SSM
18. •At low speeds, the drag force on an object moving through
a fluid is proportional to its velocity. According to Newton’s
second law, force is proportional to acceleration. As acceleration and velocity aren’t the same quantity, is there a contradiction here?
7. •Why does water stay in a bucket that is whirled around in a
vertical circle? Contrast the forces acting on the water when the
bucket is at the lowest point on the circle to when the bucket is
at the highest point on the circle. SSM
19. •James Bond leaps without a parachute from a burning
airplane flying at 15,000 ft. Ten seconds later his assistant, who
was following behind in another plane dives after him, wearing
her parachute and clinging to one for her hero. Is it possible for
her to catch up with Bond and save him?
8. •An antilock braking system (ABS) prevents wheels from
skidding while drivers stomp on the brakes in emergencies.
How far will a car with an ABS move before finally stopping,
as compared to a car without an ABS?
Multiple-Choice Questions
9. •Give two examples of two situations in which the normal force acting on an object is not equal to the object’s weight.
10. •Medical A fluidlike substance called synovial fluid lubricates the surfaces where bones meet in joints, making the coefficient of friction between bones very small. Why is the minimum
force required for moving bones in a typical knee joint different
Freed_c05_157-194hr1.indd 189
20. •If a sport utility vehicle (SUV) drives up a slope of 45°,
what must be the minimum coefficient of static friction between
the SUV’s tires and the road?
A. 1.0
B. 0.5
C. 0.7
D. 0.9
E. 0.05
8/28/12 6:24 PM
190 Chapter 5 Forces and Motion II: Applications
21. •A block of mass m slides down a rough incline with constant speed. If a similar block that has a mass of 4m were placed
on the same incline, it would
A. slide down at constant speed.
B. accelerate down the incline.
C. slowly slide down the incline and then stop.
D. accelerate down the incline with an acceleration four
times greater than that of the smaller block.
E. not move. SSM
22. •A 10-kg crate is placed on a horizontal conveyor belt moving with a constant speed. The crate does not slip. If the coefficients of friction between the crate and the belt are ms equal to
0.50 and mk equal to 0.30, what is the frictional force exerted
on the crate?
A. 98 N
B. 49 N
C. 29 N
D. 9.8 N
E. 0
23. •Biology The Escherichia coli (E. coli) bacterium propels
itself through water by means of long, thin structures called
flagella. If the force exerted by the flagella doubles, the velocity
of the bacterium
A. doubles.
B. decreases by half.
C. does not change.
D. increases by a factor of four.
E. cannot be determined without more information.
24. •A 1-kg wood ball and a 5-kg lead ball have identical sizes,
shapes, and surface characteristics. They are dropped simultaneously from a tall tower. Air resistance is present. How do their
accelerations compare?
A. The 1-kg wood ball has the larger acceleration.
B. The 5-kg lead ball has the larger acceleration.
C. The accelerations are the same.
D. The 5-kg ball accelerates at five times the acceleration
of the 1-kg ball.
E. The 1-kg ball accelerates at five times the acceleration
of the 5-kg ball.
25. •A skydiver is falling at his terminal speed. Immediately
after he opens his parachute
A. his speed will be larger than his terminal speed.
B. the drag force on the skydiver will decrease.
C. the net force on the skydiver is in the downward
direction.
D. the drag force is larger than the skydiver’s weight.
E. the net force on the skydiver is zero. SSM
26. •Two rocks that are of equal mass are tied to massless strings
and whirled in nearly horizontal circles at the same speed. One
string is twice as long as the other. What is the tension T1 in the
shorter string compared to the tension T2 in the longer one?
A. T1 = 14T2
B. T1 = 12T2
C. T1 = T2
D. T1 = 2T2
E. T1 = 4T2
27. •You are on a Ferris wheel moving in a vertical circle. When
you are at the bottom of the circle, how does the magnitude
Freed_c05_157-194hr1.indd 190
of the normal force N exerted by your seat compare to your
weight mg?
A. N = mg
B. N 7 mg
C. N 6 mg
D. N = 12mg
E. N = 2mg SSM
28. •Two rocks are tied to massless strings and whirled in
nearly horizontal circles so that the period of motion is the
same for both. One string is twice as long as the other. The
tension in the longer string is twice the tension in the shorter
one. What is the mass m1 of the rock at the end of the shorter
string compared to the mass m2 of the rock at the end of the
longer one?
A. m1 = 14m2
B. m1 = 12m2
C. m1 = m2
D. m1 = 2m2
E. m1 = 4m2
Estimation/Numerical Analysis
29. •Estimate the numerical value of the coefficient of static
friction between a car’s tires and dry pavement.
30. •Estimate the value of the coefficient of kinetic friction for
a box of books on a carpeted floor.
31. •Sports Make a rough estimate of the value of the coefficient of kinetic friction between a baseball player’s uniform and
the infield surface as he slides into second base. SSM
32. •Give a numerical estimate of the minimum radius of curvature of an unbanked on-ramp for a typical freeway.
33. •Sports Estimate the size of the coefficient of kinetic friction
between a hockey puck and the ice of a rink.
34. •(a) Estimate the magnitude of the coefficient of kinetic
friction of a book as it slides across a tabletop. (b) Estimate the
magnitude of the coefficient of static friction for the same book
on the same tabletop.
35. •Estimate the magnitude of the coefficient of kinetic friction between you and the surface of a waterslide. SSM
36. •Estimate the magnitude of the coefficient of kinetic friction for a mug of root beer as it slides across a wooden bar.
37. •The following table lists
the forces applied to a 1.00-kg
crate at particular times. The
crate begins at rest on a rough,
horizontal surface but is then
pushed with a constant force in
the horizontal direction. ­Using
the data, estimate the coefficient
of static friction and the coefficient of kinetic friction between
the crate and the surface.
t (s)
F (N)
t (s)
F (N)
0
0
0.25
8.26
0.01
1.33
0.30
7.84
0.05
3.28
0.35
5.17
0.10
8.11
0.40
5.21
0.15
8.20
0.45
5.22
0.20
8.24
0.50
5.37
Problems
5-2 The static friction force changes magnitude to offset
other applied forces
38. •What is the minimum horizontal force that will cause a
5.00-kg box to begin to slide on a horizontal surface when the
coefficient of static friction is 0.670?
8/28/12 6:24 PM
Questions and Problems 191
39. •A 7.60-kg object rests on a level floor with a coefficient of
static friction of 0.550. What minimum horizontal force will
cause the object to start sliding? SSM
cause the object to accelerate at 2.50 m>s 2? The force is applied
at an angle of u = 30.0° (Figure 5-25).
40. •Draw a free-body diagram for the situation shown in Figure 5-22. An object of mass M rests on a ramp; there is friction
between the object and the ramp. The system is in equilibrium.
M
Friction
µk
M
θ
Figure 5-25
F=?
Figure 5-22 ​​Problem 40
5-3 The kinetic friction force on a sliding object has a
constant magnitude
41. •A book is pushed across a horizontal table at a constant
speed. If the horizontal force applied to the book is equal to
one-half of the book’s weight, calculate the coefficient of ­kinetic
friction between the book and the tabletop. SSM
42. •An object on a level surface experiences a horizontal force
of 12.7 N due to kinetic friction. If the coefficient of kinetic
friction is 0.37, what is the mass of the object?
43. •A 25.0-kg crate rests on a level floor. A horizontal force of
50.0 N accelerates the crate at 1.00 m>s 2. Calculate (a) the normal force on the crate, (b) the frictional force on the crate, and (c)
the coefficient of kinetic friction between the crate and the floor.
44. •A block of mass M rests on a block of mass M1 = 5.00 kg
which is on a tabletop (Figure 5-23). A light string passes over
a frictionless peg and connects the blocks. The coefficient of
­kinetic friction mk at both surfaces equals 0.330. A force of
60.0 N pulls the upper block to the left and the lower block to
the right. The blocks are moving at a constant speed. Determine
the mass of the upper block.
5-4 Problems involving static and kinetic friction are like
any other problem with forces
47. ••A taut string connects a crate with mass M1 = 5.00 kg to
a crate with mass M2 = 12.0 kg (Figure 5-26). The coefficient of
static friction between the smaller crate and the floor is 0.573;
the coefficient of static friction between the larger crate and the
floor is 0.443. What is the minimum horizontal force F required
to start the crates in motion? SSM
M1
M2
F
Figure 5-26 ​​Problem 47
48. ••A box of mass Mbox = 2.00 kg rests on top of a crate with
mass Mcrate = 5.00 kg (Figure 5-27). The coefficient of static
friction between the box and the crate is 0.667. The coefficient
of static friction between the crate and the floor is 0.400. Calculate the minimum force F that is required to move the crate
to the right and the corresponding tension T in the rope that
connects the box to the wall when the crate is moved.
Box
T
F
M
Crate
M1
Problem 46
F
Figure 5-23
Problem 44
45. •A mop is pushed across the floor with a force F of 50.0 N at
an angle of u = 50.0° (Figure 5-24). The mass of the mop head
is 3.75 kg. Calculate the acceleration of the mop head if the coefficient of kinetic friction between the head and the floor is 0.400.
Figure 5-27 ​​Problem 48
49. •Two blocks are connected over a massless, frictionless
pulley (Figure 5-28). The mass of block 2 is 8.00 kg, and the
coefficient of kinetic friction between block 2 and the incline is
0.220. The angle u of the incline is 28.0°. Block 2 slides down
the incline at constant speed. What is the mass of block 1? SSM
F
u
µk
2
1
Figure 5-24
Problem 45
46. •If the coefficient of kinetic friction between an object with
mass M = 3.00 kg and a flat surface is 0.400, what force will
Freed_c05_157-194_st_hr2.indd 191
Friction
q
Figure 5-28
Problems 49 and 50
50. •Two blocks are connected over a massless, frictionless
pulley (Figure 5-28). The mass of block 2 is 10.0 kg, and the
4/2/13 3:08 PM
192 Chapter 5 Forces and Motion II: Applications
coefficient of kinetic friction between block 2 and the incline is
0.200. The angle u of the incline is 30.0°. If block 2 moves up
the incline at constant speed, what is the mass of block 1?
51. ••An object with mass M1 of 2.85 kg is held in place on an
inclined plane that makes an angle u of 40.0° with the horizontal (Figure 5-29). The coefficient of static friction between the
plane and the object is 0.552. A second object that has a mass
M2 of 4.75 kg is connected to the first object with a massless
string over a massless, frictionless pulley. Calculate the initial
acceleration of the system and the tension in the string once the
objects are released.
where the value of c for a 70.0-kg person with a parachute is
18.0 kg>m. (a) What is the person’s terminal velocity? (b) Without a parachute, the same person’s terminal velocity would be
about 50.0 m>s. What would be the value of the proportionality constant c in that case?
57. ••A girl rides her scooter on a hill that is inclined at 10°
with the horizontal. The combined mass of the girl and scooter
is 50.0 kg. On the way down, she coasts at a constant speed of
12.0 m>s, while experiencing a drag force that is proportional
to the square of her velocity. What force, parallel to the surface
of the hill, is required to increase her speed to 20.0 m>s? Neglect any other resistive forces.
5-6 In uniform circular motion, the net force points toward
the center of the circle
M1
M2
58. •A hockey puck that has a mass of 170 g is tied to a light
string and spun in a circle of radius 1.25 m (on frictionless ice).
If the string breaks under a tension that exceeds 5.00 N, what
is the maximum speed of the puck without breaking the string?
Static friction
Figure 5-29
u
52. •Draw free-body diagrams for the situation
shown in Figure 5-30. An
object of mass M2 rests on
a frictionless table, and an
object of mass M1 sits on
it; there is friction between
the objects. A horizontal
force Fs is applied to the
lower object as shown.
Problem 51
M1
Friction
M2
F
Frictionless
Figure 5-30 ​​Problem 52
53. ••A horizontal force F of 10.0 N is applied to a stationary
block with a mass M of 2.00 kg as shown in Figure 5-31. The
coefficient of static friction between the block and the floor is
0.750; the coefficient of kinetic friction is 0.450. Find the acceleration of the box. SSM
M
F
Friction
Figure 5-31 ​​Problem 53
5-5 An object moving through air or water experiences
a drag force
54. •Biology A single-celled animal called a paramecium propels itself quite rapidly through water by using its hairlike cilia.
A certain paramecium experiences a drag force of magnitude
Fdrag = cv2 in water, where the drag coefficient c is approximately 0.310. What propulsion force does this paramecium
generate when moving at a constant (terminal) speed v of
0.150 * 103 m>s?
55. •Biology The bacterium Escherichia coli propels itself with
long, thin structures called flagella. When its flagella exert a
force of 1.50 * 10213 N, the bacterium swims through water at
a speed of 20.0 mm>s. Find the speed of the bacterium in water
when the force exerted by its flagella is 3.00 * 10213 N.
56. ••We model the drag force of the atmosphere as proportional to the square of the speed of a falling object, Fdrag = cv2,
Freed_c05_157-194hr3.indd 192
59. •A 1500-kg truck rounds an unbanked curve on the
­highway at a speed of 20.0 m>s. If the maximum frictional
force between the surface of the road and all four of the tires
is 8000 N, calculate the minimum radius of curvature for the
curve to prevent the truck from skidding off the road. SSM
60. •A 25.0-g metal washer is tied to a 60.0-cm-long string
and whirled around in a vertical circle at a constant speed of
6.00 m>s. Calculate the tension in the string (a) when the washer is at the bottom of the circular path and (b) when it is at the
top of the path.
61. •A centrifuge spins small tubes in a circle of radius 10.0 cm
at a rate of 1200 rev>min. What is the centripetal force on a
sample that has a mass of 1.00 g? SSM
62. •Astro What centripetal force is exerted on the Moon as it
orbits about Earth at a center-to-center distance of 3.84 * 108 m
with a period of 27.4 days? What is the source of the force? The
mass of the Moon is equal to 7.35 * 1022 kg.
63. •Biology Very high-speed ultracentrifuges are useful devices
to sediment materials quickly or to separate materials. An ultracentrifuge spins a small tube in a circle of radius 10.0 cm at
60,000 rev>min. What is the centripetal force experienced by a
sample that has a mass of 0.00300 kg?
64. •At the Fermi National Accelerator Laboratory (Fermilab), a
large particle accelerator, protons are made to travel in a circular
orbit 6.3 km in circumference at a speed of nearly 3.0 * 108 m>s .
What is the centripetal acceleration of one of the protons?
65. •In the game of tetherball,
a 1.25-m rope connects a
0.750-kg ball to the top of a
vertical pole so that the ball
can spin around the pole as
shown in Figure 5-32. What
is the speed of the ball as it
rotates around the pole when
the angle u of the rope is 35.0°
with the vertical? SSM
1.25 m
u
0.75 kg
Figure 5-32 ​​Problem 65
9/17/12 4:06 PM
Questions and Problems 193
66. •What is the force that a jet pilot feels against his seat as he
completes a vertical loop that is 500 m in radius at a speed of
200 m>s? Assume his mass is 70.0 kg and that he is located at
the bottom of the loop.
67. •The radius of Earth is 6.38 * 106 m, and it completes one
revolution in 1 day. (a) What is the centripetal acceleration of
an object located on the equator? (b) What is the centripetal
acceleration of an object located at latitude 40.0° north?
68. ••A coin that has a mass of 25.0 g rests on a phonograph
turntable that rotates at 78.0 rev>min. The center of the coin
is 13.0 cm from the turntable axis. If the coin does not slip,
what is the minimum value of the coefficient of static friction
between the coin and the turntable surface?
69. •Sports In executing a windmill pitch, a fast-pitch softball
player moves her hand through a circular arc of radius 0.310
m. The 0.190-kg ball leaves her hand at 24.0 m>s. What is the
magnitude of the force exerted on the ball by her hand immediately before she releases it?
the coefficient of kinetic friction between the ski and snow is
(a) 0.100 and (b) 0.150?
75. •In a mail-sorting facility, a 2.50-kg package slides down
an inclined plane that makes an angle of 20.0° with the horizontal. The package has an initial speed of 2.00 m>s at the top
of the incline, and it slides a distance of 12.0 m. What must
the coefficient of kinetic friction between the package and the
inclined plane be so that the package reaches the bottom with
no speed? SSM
76. ••In Figure 5-34, two blocks are connected to each other
by a massless string over a massless and frictionless pulley.
The mass m1 is 6.00 kg. Assuming the coefficient of static friction ms equals 0.542 for all surfaces, find the range of values
of the mass m2 of the second block so that the system is in
equilibrium.
General Problems
70. ••A 150-kg crate rests in the bed of a truck that slows from
50.0 km>h to a stop in 12.0 s. The coefficient of static friction
between the crate and the truck bed is 0.655. (a) Will the crate
slide during the braking period? Explain your answer. (b) What
is the minimum stopping time for the truck in order to prevent
the crate from sliding?
71. ••The coefficient of static friction between a rubber tire
and dry pavement is about 0.800. Assume that a car’s engine
only turns the two rear wheels and that the weight of the car is
uniformly distributed over all four wheels. (a) What limit does
the coefficient of static friction place on the time required for
a car to accelerate from rest to 60 mph (26.8 m>s)? (b) How
can friction accelerate a car forward when friction opposes
­motion? SSM
72. ••Two blocks are connected over a massless, frictionless pulley (Figure 5-33). Block m1 has a mass of 1.00 kg and block m2
has a mass of 0.400 kg. The angle u of the incline is 30.0°. The
coefficients of static friction and kinetic friction between block
m1 and the incline are ms equal to 0.500 and mk equal to 0.400,
respectively. What is the value of the tension in the string?
m2
m1
Friction
θ
Figure 5-33
Problems 72 and 73
73. ••Two blocks are connected over a massless, frictionless
pulley (Figure 5-33). Block m1 has a mass of 1.00 kg and block
m2 has a mass of 2.00 kg. The angle u of the incline is 30.0°.
The coefficients of static friction and kinetic friction between
block m1 and the incline are ms equal to 0.500 and mk equal to
0.400. What is the acceleration of block m1?
74. •A runaway ski slides down a 250-m-long slope inclined at
37.0° with the horizontal. If the initial speed is 10.0 m>s, how
long does it take the ski to reach the bottom of the incline if
Freed_c05_157-194hr1.indd 193
m2
Static
friction
m1
60°
35°
Static
friction
Figure 5-34 Problem 76
77. ••The terminal velocity of a raindrop that is 4.00 mm in
diameter is approximately 8.50 m>s under controlled, windless conditions. The density of water is 1000 kg>m3. Recall
that the density of an object is its mass divided by its volume. (a) If we model the air drag as being proportional to
the square of the speed, Fdrag = cv2, what is the value of c?
(b) Under the same conditions as above, what would be the
terminal velocity of a raindrop that is 8.0 mm in diameter?
Try to use your answer from part (a) to solve the problem by
proportional reasoning instead of just doing the same calculation over again.
78. •Biomedical laboratories routinely use ultracentrifuges,
some of which are able to spin at 100,000 rev>min about the
central axis. The turning rotor in certain models is about 20.0
cm in diameter. At its top spin speed, what force does the bottom of the rotor exert on a 2.00-g sample that is in the rotor at
the greatest distance from the axis of spin? Would the force be
appreciably different if the sample were spun in a vertical or a
horizontal circle? Why or why not?
79. ••An amusement park ride called the Rotor debuted in
1955 in Germany. Passengers stand in the cylindrical drum of
the Rotor as it rotates around its axis. Once the Rotor reaches
its operating speed, the floor drops but the riders remain pinned
against the wall of the cylinder. Suppose the cylinder makes
25.0 rev>min and has a radius of 3.50 m. What is the coefficient of static friction between the wall of the cylinder and the
backs of the riders?
80. •An object of mass m1 undergoes constant circular motion
and is connected by a massless string through a hole in a frictionless table to a larger object of mass m2 (Figure 5-35). If the
larger object is stationary, calculate the tension in the string and
the speed of the circular motion of the smaller object. ­Assume
that the objects have masses of 0.225 and 0.125 kg and the
8/28/12 6:24 PM
194 Chapter 5 Forces and Motion II: Applications
radius R of the circular path of the smaller object is equal to
1.00 m.
R
m1
to two significant figures. (a) What is the acceleration of the
head during the collision? (b) What force (in newtons and in
pounds) does the neck exert on the head of a 75-kg person
in the collision? (As a first approximation, neglect the force
of gravity on the head.) (c) Would headrests mounted to the
backs of the car seats help protect against whiplash? Why or
why not?
83. •The wings of an airplane flying in a horizontal circle at
a speed of 680 km>h are tilted 60.0° to the horizontal (Figure 5-37). What is the radius of the circle? Assume that the
required force is provided entirely by the wings’ lift, a force
perpendicular to the surface of the wings.
m2
Figure 5-35
Problem 80
81. •An object that has a mass M hangs from a support by a
massless string of length L (Figure 5-36). The support is rotated
so that the object follows a circular path at an angle u from the
vertical as shown. The object makes N revolutions per second.
Derive an expression for the angle u in terms of M, L, N, and
any necessary physical constants. SSM
60°
L
θ
Figure 5-37 ​​Problem 83
M
84. •A curve that has a radius of 100 m is banked at an angle
of 10.0° (Figure 5-38). If a 1000-kg car navigates the curve at
65 km>h without skidding, what is the minimum coefficient of
static friction between the pavement and the tires?
Figure 5-36 Problem 81
82. •Medical Occupants of cars hit from behind, even at low
speed, often suffer serious neck injury from whiplash. During
a low-speed rear-end collision, a person’s head suddenly pivots
about the base of the neck through a 60° angle, a motion that
lasts 250 ms. The distance from the base of the neck to the center of the head is typically about 20 cm, and the head normally
comprises about 6.0% of body weight. We can model the
­motion of the head as having uniform speed over the course
of its pivot. Compute your answers to the following questions
Freed_c05_157-194hr1.indd 194
10°
Figure 5-38
​​Problem 84
8/28/12 6:25 PM