GP1 S 14 Chap 5 Problems
GP1 S 14 Chap 5 Problems
GP1 S 14 Chap 5 Problems
Exercises
Section 5.1 Using Newton's First Law:
Particles in Equilibrium
5.1. Two 25.0-N weights are suspended at opposite ends of a rope
that passes over a light, frictionless pulley. The pulley is attached
to a chain that goes to the ceiling. (a) What is the tension in the
rope? (b) What is the tension in the chain?
5.2. In Fig. 5.41 each of the suspended blocks has weight w. The
pulleys are frictionless and the ropes have negligible weight. Cal
culate, in each case, the tension T in the rope in terms of the
weight w. In each case, include the free-body diagram or diagrams
you used to determine the answer.
(b)
(c)
w
w
(a)
(b)
Exercises
169
does the surface of the ramp push on the ball? (c) What is the ten
sion in the wire?
.,aifii!lll'
5.19. Atwood's
20.0 kg
B
;;;;;g;;7
10.0 kg
-~-r--Hti7ifP
Pull
counterweight?
170
the hanging block. (d) How does the tension compare to the weight
of the hanging block?
5.22. Runway Design. A transport plane takes off from a level
landing field with two gliders in tow, one behind the other. The
mass of each glider is 700 kg, and the total resistance (air drag plus
friction with the runway) on each may be assumed constant and
equal to 2500 N. The tension in the towrope between the transport
plane and the first glider is not to exceed 12,000 N. (a) If a speed of
40 mls is required for takeoff, what minimum length of runway is
needed? (b) What is the tension in the towrope between the two
gliders while they are accelerating for the takeoff?
5.23. A 750.0-kg boulder is raised from a quarry 125 m deep by a
long uniform chain having a mass of 575 kg. This chain is of uni
form strength, but at any point it can support a maximum tension
no greater than 2.50 times its weight without breaking. (a) What is
the maximum acceleration the boulder can have and still get out of
the quarry, and (b) how long does it take to be lifted out at maxi
mum acceleration if it started from rest?
5.24. Apparent Weight. A 550-N physics student stands on a
bathroom scale in an S50-kg (including the student) elevator that is
supported by a cable. As the elevator starts moving, the scale reads
450 N. (a) Find the acceleration of the elevator (magnitude and
direction). (b) What is the acceleration if the scale reads 670 N?
(c) If the scale reads zero, should the student worry? Explain.
(d) What is the tension in the cable in parts (a) and (c)?
5.25. A physics student playing with an air hockey table (a fric
tionless surface) finds that if she gives the puck a velocity of
3.80 m/s along the length (1.75 m) of the table at one end, by the
time it has reached the other end the puck has drifted 2.50 cm to
the right but still has a velocity component along the length of
3.80 m/s. She correctly concludes that the table is not level and
correctly calculates its inclination from the given information.
What is the angle of inclination?
5.26. A 2540-kg test rocket is launched vertically from the launch
pad. Its fuel (of negligible mass) provides a thrust force so that its
vertical velocity as a function of time is given by v (t) =
At + Bt 2, where A and B are constants and time is measured from
the instant the fuel is ignited. At the instant of ignition, the rocket
has an upward acceleration of 1.50 ml S2 and 1.00 s later an upward
velocity of 2.00 m/s. (a) Determine A and B, including their SI
units. (b) At 4.00 s after fuel ignition, what is the acceleration of
the rocket, and (c) what thrust force does the burning fuel exert on
it, assume no air resistance?
the thrust in newtons and as a
mUltiple of the rocket's weight. (d) What was the initial thrust due
to the fuel?
Exercises
171
J'vk' The crat~ are pulled to the right at constant velocity by a hori
zontal force F. In ~rms of mA, m B, and J'vk. calculate (a) the magni
T
2.50
m
~_~J
exert to accomplish this? (b) What are the magnitude and direction
of the friction force on the upper box?
5.34. Stopping Distance. (a) If the coefficient of kinetic friction
between tires and dry pavement is 0.80, what is the shortest dis
tance in which you can stop an automobile by locking the brakes
when traveling at 28.7 mls (about 65 rni/h)? (b) On wet pavement
the coefficient of kinetic friction may be only 0.25. How fast
should you drive on wet pavement in order to be able to stop in the
same distance as in part (a)? (Note: Locking the brakes is not the
safest way to stop.)
5.35. Coefficient of Friction. A clean brass washer slides along
a horizontal clean steel surface until it stops. Using the values from
Table 5.1, how many times farther would it slide with the same ini
tial speed if the washer were Teflon-coated?
5.36. Consider the system shown in Fig. 5.54. Block A weighs
45.0 N and block B weighs 25.0 N. Once block B is set into down
ward motion, it descends at a constant speed. (a) Calculate the
coefficient of kinetic friction between block A and the tabletop.
(b) A cat, also of weight 45.0 N, falls asleep on top of block A. If
block B is now set into downward motion, what is its acceleration
(magnitude and direction)?
tude of the force F and (b) the tension in the rope connecting the
blocks. Include the free-body diagram or diagrams you used to
determine each answer.
5.38. Rolling Friction. Two bicycle tires are set rolling with the
same initial speed of 3.50 mls on a long, straight road, and the dis
tance each travels before its speed is reduced by half is measured.
One tire is inflated to a pressure of 40 psi and goes 18.1 m; the
other is at 105 psi and goes 92.9 m. What is the coefficient of
rolling friction J'vr for each? Assume that the net horizontal force is
due to rolling friction only.
5.39. Wheels. You find that it takes a horizontal force of 160 N
to slide a box along the surface of a level floor at constant speed.
The coefficient of static friction is 0.52, and the coefficient of
kinetic friction is 0.47. If you place the box on a dolly of mass
5.3 kg and with coefficient of rolling friction 0.018, what horizon
tal acceleration would that 160-N force provide?
5.40. You find it takes 200 N of horizontal force to move an empty
pickup truck along a level road at a speed of 2.4 m/s. You then
load the pickup and pump up its tires so that its total weight
increases by 42% while the coefficient of rolling friction decreases
by 19%. Now what horizontal force will you need to move the
pickup along the same road at the same speed? The speed is low
enough that you can ignore air resistance.
5.41. As shown in Fig. 5.54, block A (mass 2.25 kg) rests on a
tabletop. It is connected by a horizontal cord passing over a light,
frictionless pulley to a hanging block B (mass 1.30 kg). The coeffi
cient of kinetic friction between block A and the tabletop is 0.450.
After the blocks are released from rest, find (a) the speed of each
block after moving 3.00 cm and (b) the tension in the cord. Include
the free-body diagram or diagrams you used to determine the
answers.
5.42. A 25.0-kg box of textbooks rests on a loading ramp that
makes an angle a with the horizontal. The coefficient of kinetic
friction is 0.25, and the coefficient of static friction is 0.35. (a) As
the angle a is increased, find the minimum angle at which the box
starts to slip. (b) At this angle, find the acceleration once the box
has begun to move. (c) At this angle, how fast will the box be mov
ing after it has slid 5.0 m along the loading ramp?
5.43. A large crate with mass m rests on a horizontal floor. The
coefficients of friction between the crate and the floor are J'vs and
J'vk' A woman pushes downward at an angle (J below the horizontal
on the crate with a force F. (a) What magnitude of force F is
required to keep the crate moving at constant velocity? (b) If J'vs is
greater than some critical value, the woman cannot start the crate
moving no matter how hard she pushes. Calculate this critical
value of J'vS'
5.44. A box with mass m is dragged across a level floor having a
coefficient of kinetic friction J'vk by a rope that is pulled upward at
an angle (J above the horizontal with a force of magnitude F. (a) In
terms of m, J'vk' (J, and g, obtain an expression for the magnitude of
force required to move the box with constant speed. (b) Knowing
that you are studying physics, a CPR instructor asks you how
much force it would take to slide a 90-kg patient across a floor at
constant speed by pulling on him at an angle of 25 above the hor
izontaL By dragging some weights wrapped in an old pair of pants
down the hall with a spring balance, you find that J'vk 0.35. Use
the result of part (a) to answer the instructor's question.
172
,
f
. Ii
Ii
II
II
i
a point 3.00 m from the central shaft. (a) Find the time of one rev
olution of the swing if the cable supporting a seat makes an angle
of 30.0 with the verticaL (b) Does the angle depend on the weight
of the passenger for a given rate of revolution?
5.53. In another version of Figure 5.58 Exercise 5.53.
the "Giant Swing" (see Exer
cise 5.52), the seat is con
nected to two cables as shown
in Fig. 5.58, one of which is
horizontaL The seat swings in
a horizontal circle at a rate of
32.0 rpm (rev/min). If the
seat weighs 255 N and a 825-N
person is sitting in it, find the
tension in each cable.
5.54. A small button placed on
a horizontal rotating platform
with diameter 0.320 m will
revolve with the platform when it is brought up to a speed of
40.0 rev/min, provided the button is no more than 0.150 m from
the axis. (a) What is the coefficient of static friction between the
button and the platfoml? (b) How far from the axis can the button
be placed, without slipping, if the platform rotates at 60.0 rev/min?
5.55. Rotating Space Stations. One problem for humans living
in outer space is that they are apparently weightless. One way around
this problem is to design a space station that spins about its center at
a constant rate. This creates "artificial gravity" at the outside rim of
the station. (a) If the diameter of the space station is 800 m, how
many revolutions per minute are needed for the "artificial gravity"
acceleration to be 9.80 m/s 2 ? (b) If the space station is a waiting
area for travelers going to Mars, it might be desirable to simulate
the acceleration due to gravity on the Martian surface (3.70 m/s2 ).
How many revolutions per minute are needed in this case?
5.56. The Cosmoc1ock 21 Ferris wheel in Yokohama City, Japan,
has a diameter of 100 m. Its name comes from its 60 arms, each of
which can function as a second hand (so that it makes one revolu
tion every 60.0 s). (a) Find the speed of the passengers when the
Ferris wheel is rotating at this rate. (b) A passenger weighs 882 N
at the weight-guessing booth on the ground. What is his apparent
weight at the highest and at the lowest point on the Ferris wheel?
(c) What would be the time for one revolution if the passenger's
apparent weight at the highest point were zero? (d) What then
would be the passenger's apparent weight at the lowest point?
5.57. An airplane flies in a loop (a circular path in a vertical plane)
of radius 150 m. The pilot's head always points toward the center
of the loop. The speed of the airplane is not constant; the airplane
goes slowest at the top of the loop and fastest at the bottom. (a) At
0
5.46. Starting from Eq. (5.10), derive Eqs. (5.11) and (5.12).
5.47. (a) In Example 5.19 (Section 5.3), what value of D is
required to make v, = 42 m/s for the skydiver? (b) If the sky
diver's daughter, whose mass is 45 kg, is falling through the air
and has the same D (0.25 kg 1m ) as her father, what is the daugh
ter's terminal speed?
5.48. You throw a baseball straight up. The drag force is propor
tional to v2 In terms of g, what is the y-component of the ball's
acceleration when its speed is half its terminal speed and (a) it is
moving up? (b) It is moving back down?
Problems
the top of the loop, the pilot feels weightless. What is the speed of
the airplane at this point? (b) At the bottom of the loop, the speed
of the airplane is 280 kID/h. What is the apparent weight of the
pilot at this point? His true weight is 700 N.
5.58. A 50.0-kg stunt pilot who has been diving her airplane verti
cally pulls out of the dive by changing her course to a circle in a
vertical plane. (a) If the plane's speed at the lowest point of the cir
cle is 95.0 mis, what is the minimum radius of the circle for the
acceleration at this point not to exceed 4.008? (b) What is the
apparent weight of the pilot at the lowest point of the pullout?
5.59. Stay Dry! You tie a cord to a pail of water, and you swing
the pail in a vertical circle of radius 0.600 m. What minimum
speed must you give the pail at the highest point of the circle if no
water is to spill from it?
5.60. A bowling ball weighing 71.2 N (16.0 lb) is attached to the
ceiling by a 3.80-m rope. The ball is pulled to one side and
released; it then swings back and forth as a pendulum. As the rope
swings through the vertical, the speed of the bowling ball is
4.20 m/s. (a) What is the acceleration of the bowling ball, in mag
nitude and direction, at this instant? (b) What is the tension in the
rope at this instant?
Problems
173
objects in the problem that you can safely ignore their mass. But if
the rope is the only object in the problem; then clearly you cannot
ignore its mass. For example, suppose we have a clothesline
attached to two poles (Fig. 5.61). The clothesline has a mass M,
and each end makes an angle ewith the horizontal. What are (a) the
tension at the ends of the clothesline and (b) the tension at the low
est point? (c) Why can't we have () = O? (See Discussion Ques
tion Q5.3.) (d) Discuss your results for parts (a) and (b) in the limit
that e ~ 90. The curve of the clothesline, or of any flexible cable
hanging under its own weight, is called a catenary. [For a more
advanced treatment of this curve, see K. R. Symon, Mechanics, 3rd
ed. (Reading, MA: Addison-Wesley, 1971), pp. 237-241.]
Figure 5.61 Problem 5.63.
mum value of the hanging weight that these ropes can safely sup
port. You can ignore the weight of the ropes and the steel cable.
chain, and the lower pulley is attached to the weight by another chain.
the force F if the weight is lifted at constant speed. Include the free
Assume that the rope, pulleys, and chains all have negligible weights.
174
(a)
(b)
100
a/g
50
o~------~-----------------o
0.5
1.0
1.5
Time (ms)
5.70. A 2S,000-kg rocket blasts off vertically from the earth's sur
face with a constant acceleration. During the motion considered in
the problem, assume that g remains constant (see Chapter 12).
Inside the rocket, a 15.0-N instrument hangs from a wire that can
Problems
decrease, or remain constant? Explain. (b) Let rnA = 2.00 kg,
ms OAOO kg, rnrope = 0.160 kg, and L
1.00 m. If there is fric
tion between block A and the tabletop, with ILk == 0.200 and
ILs
0.250, find the minimum value of the distance d such that
the blocks will start to move if they are initially at rest. (c) Repeat
part (b) for the case rnrope == 0.040 kg. Will the blocks move in this
case?
5.78. If the coefficient of static friction between a table and a uni
form massive rope is IL" what fraction of the rope can hang over
the edge of the table without the rope sliding?
5.79. A 30.0-kg packing case is initially at rest on the floor of a
1500-kg pickup truck. The coefficient of static friction between the
case and the truck floor is 0.30, and the coefficient of kinetic fric
tion is 0.20. Before each acceleration given below, the truck is
traveling due north at constant speed. Find the magnitude and
direction of the friction force acting on the case (a) when the truck
accelerates at 2.20 m/s2 northward and (b) when it accelerates at
3.40 m/s 2 southward.
5.80. Traffic Court. You are called as an expert witness in the
trial of a traffic violation. The facts are these: A driver slammed on
his brakes and came to a stop with constant acceleration. Measure
ments of his tires and the skid marks on the pavement indicate that
he locked his car's wheels, the car traveled 192 ft before stopping,
and the coefficient of kinetic friction between the road and his tires
was 0.750. The charge is that he was speeding in a 45-mi/h zone.
He pleads innocent. What is your conclusion, guilty or innocent?
How fast was he going when he hit his brakes?
5.81. Two identical 15.0-kg balls, each 25.0 cm in diameter, are
suspended by two 35.0-cm wires as shown in Fig. 5.67. The entire
apparatus is supported by a single 18.0-cm wire, and the surfaces
of the balls are perlectly smooth. (a) Find the tension in each of the
three wires. (b) How hard does each ball push on the other one?
175
5.84. You are part of a design tearn for future exploration of the
planet Mars, where g = 3.7 m/52 An explorer is to step out of a
survey vehicle traveling horizontally at 33 m/s when it is 1200 m
above the surface and then fall freely for 20 s. At that time, a
portable advanced propulsion system (PAPS) is to exert a constant
force that will decrease the explorer's speed to zero at the instant
she touches the surface. The total mass (explorer, suit, equipment,
and PAPS) is 150 kg. Assume the change in mass of the PAPS to
be negligible. Find the horizontal and vertical components of the
force the PAPS must exert, and for what interval of time the PAPS
must exert it. You can ignore air resistance.
5.85. Block A in Fig. 5.69 has a mass of 4.00 kg, and block B has
mass 12.0 kg. The coefficient of kinetic friction between block B
and the horizontal surface is 0.25. (a) What is the mass of block C
if block B is moving to the right and speeding up with an accelera
tion 2.00 m/s 2 ? (b) What is the tension in each cord when block B
has this acceleration?
176
5.87. In terms of ml, /112, and g, find the accelerations of each block
in Fig. 5.7l. There is no friction anywhere in the system.
Figure 5.71 Problem 5.87.
5.88. Block B, with mass 5.00 kg, rests on block A, with mass
8.00 kg, which in turn is on a horizontal tabletop (Fig. 5.72). There
is no friction between block A and the tabletop, but the coefficient
of static friction between block A and block B is 0.750. A light
string attached to block A passes over a frictionless, massless pul
ley, and block C is suspended from the other end of the string.
What is the largest mass that block C can have so that blocks A and
B still slide together when the system is released from rest?
Figure 5.72 Problem 5.88.
c
5.89. Two objects with masses 5.00 kg and 2.00 kg hang 0.600 m
above the floor from the ends of a cord 6.00 m long passing over a
frictionless pulley. Both objects start from rest. Find the maximum
height reached by the 2.00-kg object.
5.90. Friction in an Elevator. You are riding in an elevator on
the way to the 18th floor of your dormitory. The elevator is accel
erating upward with a = 1.90 m/s 2 Beside you is the box contain
ing your new computer; the box and its contents have a total mass
of 28.0 kg. While the elevator is accelerating upward, you push
horizontally on the box to slide it at constant speed toward the ele
vator door. If the coefficient of kinetic friction between the box and
the elevator floor is fLk = 0.32, what magnitude of force must you
apply?
5.91. A block is placed against the vertical front of a cart as shown
in Fig. 5.73. What acceleration must the cart have so that block A
does not fall? The coefficient of static friction between the block
and the cart is fL,. How would an observer on the cart describe the
behavior of the block?
Figure 5.73 Problem 5.91.
5.94. Accelerometer.
The system shown in Fig. 5.76 can be
used to measure the acceleration of the system. An observer riding
on the platform measures the angle (J that the thread supporting the
light ball makes with the vertical. There is no friction anywhere.
(a) How is (J related to the acceleration of the system? (b) If
ml = 250 kg and m2
1250 kg, what is (J? (c) If you can vary /111
and /112, what is the largest angle e you could achieve? Explain
how you need to adjust ml and /112 to do this.
Figure 5.76 Problem 5.94.
Problems
5.95. Banked Curve I. A curve with a 120-m radius on a level
road is banked at the correct angle for a speed of 20 m/s. If an
automobile rounds this curve at 30 mIs, what is the minimum
coefficient of static friction needed between tires and road to pre
vent skidding?
5.96. Banked Curve II. Consider a wet roadway banked as in
Example 5.23 (Section 5.4), where there is a coefficient of static
friction of 0.30 and a coefficient of kinetic friction of 0.25 between
the tires and the roadway. The radius of the curve is R = 50 m.
(a) If the banking angle is f3 = 25, what is the maximum speed
the automobile can have before sliding up the banking? (b) What
is the minimum speed the automobile can have before sliding down
the banking?
5.97. Maximum Safe Speed. As you travel every day to campus,
the road makes a large tum that is approximately an arc of a circle.
You notice the warning sign at the start of the tum, asking for a max
imum speed of 55 mi/h. You also notice that in the curved portion
the road is level-that is, not banked at all. On a dry day with very
little traffic, you enter the tum at a constant speed of SO milh and
feel that the car may skid if you do not slow down quickly. You con
clude that your speed is at the limit of safety for this curve and you
slow down. However, you remember reading that on dry pavement
new tires have an average coefficient of static friction of about 0.76,
while under the worst winter driving conditions, you may encounter
wet ice for which the coefficient of static friction can be as low as
0.20. Wet ice is not unheard of on this road, so you ask yourself
whether the speed limit for the tum on the roadside warning sign is
for the worst-case scenario. (a) Estimate the radius of the curve
from your SO-mi/h experience in the dry tum. (b) Use this estimate
to find the maximum speed limit in the tum under the worst wet-ice
conditions. How does this compare with the speed limit on the sign?
Is the sign misleading drivers? (c) On a rainy day, the coefficient of
static friction would be about 0.37. What is the maximum safe speed
for the tum when the road is wet? Does your answer help you under
stand the maximum-speed sign?
5.98. You are riding in a school bus. As the bus rounds a flat curve
at constant speed, a lunch box with mass 0.500 kg, suspended from
the ceiling of the bus by a string I.S0 m long, is found to hang at
rest relative to the bus when the string makes an angle of 30.0
with the vertical. In this position the lunch box is 50.0 m from the
center of curvature of the curve.
What is the speed v of the bus?
Figure 5.77 Problem 5.99.
5.99. The
Monkey
and
Bananas Problem. A 20-kg
monkey has a firm hold on a
light rope that passes over a
frictionless pulley and is
attached to a 20-kg bunch of
bananas (Fig. 5.77). The monkey
looks up, sees the bananas, and
starts to climb the rope to get
them. (a) As the monkey climbs,
do the bananas move up, down,
or remain at rest? (b) As the
monkey climbs, does the dis
tance between the monkey and
the bananas decrease, increase,
or remain constant? (c) The
monkey releases her hold on the
rope. What happens to the dis
tance between the monkey and
the bananas while she is falling?
171
(d) Before reaching the ground, the monkey grabs the rope to stop
her fall. What do the bananas do?
.
5.100. You throw a rock downward into water with a speed of
3mglk, where k is the coefficient in
(5.7). Assume that the rela
tionship between fluid resistance and speed is as given in Eq. (5.7),
and calculate the speed of the rock as a function of time.
5.101. A rock with mass m = 3.00 kg falls from rest in a viscous
medium. The rock is acted on by a net constant downward force of
IS.0 N (a combination of gravity and the buoyant force exerted by
ku, where v is the
the medium) and by a fluid resistance force f
speed in mls and k = 2.20 N . slm (see Section 5.3). (a) Find the
initial acceleration ao. (b) Find the acceleration when the speed is
3.00 m/s. (c) Find the speed when the acceleration equals O.lao.
(d) Find the terminal speed v,, (e) Find the coordinate, speed, and
acceleration 2.00 s after the start of the motion. (f) Find the time
required to reach a speed 0.9v,.
5.102. A rock with mass m slides with initial velocity Vo on a hori
zontal surface. A retarding force FR that the surface exerts on the
rock is proportional to the square root of the instantaneous veloc
ity of the rock (FR = -ku l /2). (a) Find expressions for the veloc
ity and position of the rock as a function of time. (b) In terms of
m, k, and vo, at what time will the rock come to rest? (c) In terms
of m, k, and vo, what is the distance of the rock from its starting
point when it comes to rest?
5.103. A fluid exerts an upward buoyancy force on an object
immersed in it. In the derivation of Eq. (5.9) the buoyancy force
exerted on an object by the fluid was ignored. But in some situa
tions, where the density of the object is not much greater than the
density of the fluid, you cannot ignore the buoyancy force. For a
plastic sphere falling in water, you calculate the terminal speed to
be 0.36 mls when you ignore buoyancy, but you measure it to be
0.24 m/s. The buoyancy force is what fraction of the weight?
5.104. The 4.00-kg block in Figure 5.78 Problem 5.104.
5.7S is attached to a verti
cal rod by means of two
strings. When the system
totates about the axis of the
1.25 m
rod, the strings are extended as
shown in the diagram and the
4.00 kg
tension in the upper string is 2.00m
SO.O N. (a) What is the tension
1.25 ill
in the lower cord? (b) How
many revolutions per minute
does the system make? (c) Find
the number of revolutions per
minute at which the lower cord
just goes slack. (d) Explain
what happens if the number of revolutions per minute is less than
in part (c).
5.105. Equation (5.10) applies to the case where the initial velocity
is zero. (a) Derive the corresponding equation for vy(t) when the
falling object has an initial downward velocity with magnitude Va.
(b) For the case where va < v,, sketch a graph of Vy as a function
of t and label v, on your graph. (c) Repeat part (b) for the case
where Vo > v,. (d) Discuss what your result says about vy(t) when
Va
= v,,
178
(b) When the effects of fluid resistance are included, what are the
answers to the questions in part (a)?
5.107. You observe a l350-kg sports car rolling along flat pave
ment in a straight line. The only horizontal forces acting on it are a
constant rolling friction and air resistance (proportional to the
square of its speed). You take the following data during a time
interval of 25 s: When its speed is 32 mis, the car slows down at a
rate of -0,42 m/s2 , and when its speed is decreased to 24 mis, it
slows down at -0.30 m/s2 . (a) Find the coefficient of rolling fric
tion and the air
constant D. (b) At what constant speed will
this car move down an incline that makes a 2.2" angle with the
horizontal? (c) How is the constant speed for an incline of angle f3
related to the terminal speed of this sports car if the car drops off a
high cliff? Assume that in both cases the air resistance force is pro
portional to the square of the speed, and the air drag constant is the
same.
5.10B. A 70-kg person rides in a 30-kg cart moving at 12 m/s at the
top of a hill that is in the shape of an arc of a circle with a radius of
40 m. (a) What is the apparent weight of the person as the cart
passes over the top of the hill? (b) Determine the maximum speed
that the cart may travel at the top of the hill without losing contact
with the surface. Does your answer depend on the mass of the cart
or the mass of the person? Explain.
5.109. Merry-Go-Round. One December identical twins Jena
and Jackie are playing on a large merry-go-round (a disk mounted
parallel to the ground, on a vertical axle through its center) in their
school playground in northern Minnesota. Each twin has mass
30.0
The icy coating on the merry-go-round surface makes it
frictionless. The merry-go-round revolves at a constant rate as the
twins ride on it. Jena, sitting 1.80 m from the center of the merry
go-round, must hold on to one of the metal posts attached to the
merry-go-round with a horizontal force of 60.0 N to keep from
sliding off. Jackie is sitting at the edge, 3.60 m from the center.
(a) With what horizontal force must Jackie hold on to keep from
falling off? (b) If Jackie falls off, what will be her horizontal veloc
ity when she becomes airborne?
5.110. A passenger with mass 85 kg rides in a Ferris wheel like that
in Example 5.24 (Section 5,4). The seats travel in a circle of radius
35 m. The Ferris wheel rotates at constant speed and makes one
complete revolution every 25 s. Calculate the magnitude and direc
tion of the net force exerted on the passenger by the seat when she
is (a) one-quarter revolution past her lowest point and (b) one
quarter revolution past her highest point.
5.111. On the ride "Spindletop" at the amusement
Six Flags
Over Texas, people stood against the inner wall of a hollow verti
cal cylinder with radius 2.5 m. The cylinder started to rotate, and
when it reached a constant rotation rate of 0.60 revIs, the floor on
which people were standing dropped about 0.5 m. The people
remained pinned against the wall. (a) Draw a force diagram for a
person on this ride, after the floor has dropped. (b) What minimum
coefficient of static friction is required if the person on the ride is
not to slide downward to the new position of the floor? (c) Does
your answer in part (b) depend on the mass of the passenger?
(Note: When the ride is over, the cylinder is slowly brought to rest.
As it slows down, people slide down the walls to the floor.)
5.112. A physics major is working to pay his college tuition by per
forming in a traveling carnival. He rides a motorcycle inside a hol
low transparent plastic sphere. After gaining sufficient speed, he
travels in a vertical circle with a radius of l3.0 m. The physics
major has mass 70.0 kg, and his motorcycle has mass 40.0 kg.
(a) What minimum speed must he have at the top of the circle if
the tires of the motorcycle are not to lose contact with the sphere?
(b) At the bottom of the circle, his speed is twice the value calcu
lated in part (a). What is the magnitude of the normal force exerted
on the motorcycle by the sphere at this point?
5.113. Ulterior Motives. You are driving a classic 1954 Nash
Ambassador with a friend who is sitting to your right on the pas
senger side of the front seat. The Ambassador has flat bench seats.
You would like to be closer to your friend and decide to use phYSics
to achieve your romantic goal by making a quick tum. (a) Which
way (to the left or to the right) should you tum the car to get your
friend to slide closer to you? (b) If the coefficient of static friction
between your friend and the car seat is 0.35, and you keep driving
at a constant speed of 20 mIs, what is the maximum radius you
could make your turn and still have your friend slide your way?
5.114. A small block with mass m rests on a frictionless horizontal
tabletop a distance r from a hole in the center of the table
(Fig. 5.79). A string tied to the small block passes down through
the hole, and a larger block with mass M is suspended from the
free end of the string. The small block is set into uniform circular
motion with radius r and speed v. What must v be if the large
block is to remain motionless when released?
Figure 5.79 Problem 5.114.
179
Challenge Problems
5.116. A model airplane with mass 2.20 kg moves in the xy-plane
such that its x- and y-coordinates vary in time according to
x(t) = a - f3t 3 and y(t) = yt - &2, where a 1.50 m, f3
0.120 m/s3, y
3.00 mIs, and 0 = 1.00 m/s 2 (a) Calculate the
x- and y-components of the net force on the plane as functions of
time. (b) Sketch the trajectory of the airplane between t 0 and
t = 3.00 s, and draw on your sketch vectors showing the net force
on the airplane at t 0, t = 1.00 s, t
2.00 s, and t = 3.00 s. For
each of these times, relate the direction of the net force to the
direction that the airplane is turning, and to whether the airplane is
speeding up or slowing down (or neither). (c) What are the magni
tude and direction of the net force at t = 3.00 s?
5.117. A particle moves on a frictionless surface along a path as
shown in Fig. 5.81. (The figure gives a view looking down on the
surface.) The particle is initially at rest at point A and then begins
to move toward B as it gains speed at a constant rate. From B to C,
the particle moves along a circular path at a constant speed. The
speed remains constant along the straight-line path from C to D.
From D to E, the particle moves along a circular path, but now its
speed is decreasing at a constant rate. The speed continues to
decrease at a constant rate as the particle moves from E to F; the
particle comes to a halt at F. (The time intervals between the
marked points are not equal.) At each point marked with a dot,
draw arrows to represent the velocity, the acceleration, and the net
force acting on the particle. Draw longer or shorter arrows to rep
resent vectors of larger or smaller magnitude.
c_t--=::?
\ 1
\
I~m
~-
\
\\
\.J!.-j
\f
Challenge Problems
{3
(a)
+ ...... ,
B
\
't C
TD
(b)
;
FE.;
f---
- --t-......
180
dx
1
= - arctanh
Q
your expressions give for the special case of In, = 1n2 and m3 =
+ 1n2? Is this sensible?
5.126. The masses of blocks A and B in
5.86 are 20.0 kg and
10.0 kg, respectively. The blocks are initially at rest on the floor
and are connected by a massless string passing over a massless
and frictionless pulley. An upward force F is applied to the pul
ley. Find the accelerations aA of block A and aBof block B when F
is (a) 124 N; (b) 294 N; (c) 424 N.
In I
1Ix)
\Q
where
tanh (x )
eX-e- X
eX
+e
ell:-I
ell:
20.0 kg
10.0 kg
A-IO
4.27 b) yes
4.31 b) 142 N
b) 840 N
d) 25.0 N
4.47 b) 79.6 N
d) m(g - lal)
c) 4532 N. 6.16mg
4.53 a) w b) 0 c) wl2
4.55 b) 1390 N
4.59 -6mBt
5.87
a,
= 2m,81(4m,
a,
+ m,);
m,g/(4m, +
mJ
5.91
g/l1-,
5.93 b) 0.450
5.95 0.34
c) 25 m/s. 56 milh
f) 3.14 s
5.103 1/3
is (sintJ - 0.015costJ)'"
5.111 b) 0.28 c) no
5.119 T...,
Tmin = 21T
Chapter 5
5.1
5.3
5.5
5.7
5.9
5.11
5.13
5.15
5.17
5.19
5.21
5.23
5.25
5.29
5.31
5.33
5.35
5.37
5.39
5.41
5.43
5.45
5.47
5.49
5.51
5.53
5.55
5.57
5.59
5.61
5.63
5.65
5.J21
a) 25.0N b) 50.0N
48
4.10 X 101 N
B: 3.35w; C: w
a) 337 N b) 343 N
a) 470 N b) 163 N
b) 1.22ntg c) 0.70mg
a) 4610
470g b) 9.70 X 10' N. 471w
c) 18.7
b) 2.96
c) 191 N; more than the bricks,
c) 1.37 kg d) T
0.745w
b) 2.50
a) 0.832 mis' b) 17.3 s
1.38'
a) 22 N b) 3.1 m
(li) 4.97
a) 57.IN b) 146N.uptheramp
II times farther
3.82
a) 0.218 m/s b) 11.7 N
2.36 X 10" N
2.43 m/s
b) m,(sina
I1-kCosa)
c) nt, (sina
/.<,cosa) < m, <
c) IJ
5.125 a)
a,
4mlm2
b) an = -a,
c)
a,
e) TA
mis'
f) Tc
d)
g)
+ m21113
14m/In:!
81
\ 4mlm2
mis',
ms
mis'
3m21nJ
m21n3
+ tn3fnl
+ 1n3ml
Q,
6.67
6.69
6.71
6.73
6.75
6.77
6.79
6.81
6.83
6.85
6.87
6.89
6.91
6.93
6.95
6.97
6.99
6.101
6.103
4mjn12
0, Tc = 2m,8.
T, = m,g; yes
5.127 cos ',/3
7.11
7.13
mis'
mis'
mB
\rnA
+ m,op,dIL ) .
111B
+ ntrope
; mcreases
b) 0.63 m
5.83 2.52 N
in right-hand cord
Chapter 6
6.1
6.3
6.5
6.7
6.9
6.11
6.13
6.15
6.17
6.19
6.21
6.23
6.25
6.27
6.29
6.31
6.33
6.35
6.37
6.39
6.41
6.43
6.45
6.47
6.49
6.51
e) zero
a) -1750 J b) no
(ii) -25.1 J
a) 1.0 X 10'6 J b) about 2 times greater
a) 42.85V b) 1836K
a) 9D b) DI3
32.0 N
.).4.48
b) 3.61 mls
same
aJ 2.8
b) 3.5 m/s
8.5 cm
a) 1.76 bl 0.67.m/s
e) -1.0 J
743 W, 0.995 hp
a) 85 b) 23
m/s
m/s
mls
mls
7.15
7.17
7.19
7.21
7.23
7.25
7.27
7.29
7.31
I)
-;
Xz
iMu'
Chapter 7
+ m2m3 + m3ntl
= a, = G, = an
XI
0.786
1.5 m
c) 3.99 kW
3.6 h
a) 2.4 MW b) 61 MW c) 6.0 MW
a)
b) 6.1 m/s cJ 3.9 mls
7.1
7.3
7.5
7.7
7.9
---=--'--"-'''--
(1
I;
XII
+ m3mJ)
m:.;ntj
d) -203 J
J \ negative b) k -
\X2
+ m2m3 - 3m31n1)
Q, = g 4ntlm2
__---''----=-''----=--'
(
4m[m'Z + m znt3 + 1113ml
~
rI
6.65 a) k'
mis'
5.71 1040 N
5.75 0.40
5.77 aj 8
5.123 a) F
6.55 877 J
WoO<
aJ 24.0
b) 24.0
c) part (b)
mls
mls
2.5 mls
friction d) 5.0 N
-5400J
e) a
same
a) 80.0 J b) 5.00 J
(li) xo/v'2
a) 6.32 Cm b) 12 em
:to.092 m
d) nonconservative
e)
7.35 c) attracts
(12a/r'"')
(6blr7)
7.37 a) F(r)
b) (2a/b)''''; stable c)
d) a = 6.68 X 10- 138 1 m".
b = 6.41 X 1O~" J . mb
7.43 0.41
section b) -78.4 J
7.51 0.602 m
b'/4a