Download Chapter 9 - Churchill High School

Chapter 9: Periodic Motion
Assignment
Centripetal Force (a)
1
The royal antelope of western Africa has an average mass
of only 3.2 kg. If this antelope runs with a tangential speed
of 9.1 m/s in a circle with a radius of 30.0 m, how large is
the force that maintains this circular motion?
6
ANSWERS
ANSWERS
1.9 x 1020 N
8.8 N
2
Mata Jagdamba of India had the longest hair in 1994; it was
measured to be 4.23 m long. Suppose Jagdamba conducts
experiments with her hair. First, she determines that one
hair can support a mass of 25 g. She then attaches a smaller
mass to the same hair and swings it in the horizontal plane.
If the hair breaks when the tangential speed of the mass
reaches 8.1 m/s, how large is the mass?
7
Pat Kinch cycled 75.57 km in 1.000 h. Suppose the entire
distance was made by traveling once around a large circular
path. If Kinch's mass is 72 kg, estimate the average force
that maintains her circular motion.
ANSWERS
2.6 N
ANSWERS
15.8 g
8
3
The moon (mass = 7.36 x 1022 kg) orbits Earth at a range of
3.84 x 105 km with a period of approximately 28 days.
Determine the force that maintains the circular motion of
the moon.
A mass m = .2 kg on a frictionless table is attached to a
hanging mass M = 3 kg by a cord through a hole in the
table shown in the figure. Find the speed with which m
must spin for M to stay at rest.
An 86.5 kg bicyclist is riding at a linear speed of 13.2 m/s
around a circular track with a radius of 40.0 m. Find the
magnitude of the force that maintains the bike's circular
motion.
ANSWERS
377 N
9
An airplane is flying in a horizontal circle at a speed of 105
m/s. The 80.0 kg pilot does not want his centripetal
acceleration to exceed 7.00 times free fall acceleration.
(a) What is the radius of the planes circle?
(b) What is the force on the piolet?
ANSWERS
(a) 161 m
(b) 5.49 x 103 N
ANSWERS
10
4
A 70.5 kg pilot is flying a small plane at 30.0 m/s in a
circular path with a radius of 100.0 m. Find the magnitude
of the force that maintains the circular motion of the pilot.
ANSWERS
635 N
5
An astronaut who weighs 735 N on Earth is at the rim of a
cylindrical space station with a 73 m radius. The space
station is rotating at an angular speed of 3.5 rpm. Evaluate
the force that maintains the circular motion of the astronaut.
ANSWERS
735 N
Gregg Reid of Atlanta, Georgia built a motorcycle that was
over 4.5 m long and had a mass of 235 kg. If it takes Reid
12.8 s to ride his motorcycle in a circle that has a radius of
25.0 m, how great is the force that holds the motorcycle in a
circular path? Assume Reid's mass is 72 kg.
ANSWERS
1.85 x 103 N
Chapter 9: Periodic Motion
Assignment
Centripetal Force (a)
11
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock tied to a post and
moving in a circle at constant speed on a frictionless
horizontal surface.
14
A puck of mass 0.025 kg is tied to a string and allowed to
revolve in a circle of radius 1.0 m on a frictionless
horizontal surface. The other end of the string passes
through a hole in the center of the surface, and a mass of
1.0 kg is tied to it, as shown in the figure below. The
suspended mass remains in equilibrium while the puck
revolves on the surface.
(a) What is the magnitude of the force that maintains
circular motion acting on the puck?
(b) What is the linear speed of the puck?
ANSWERS
12
A 3-kg body is acted on by a single force F perpendicular to
the velocity of the body. The body travels in a circle of
radius 2 m. It makes one complete revolution every 3s.
(a) What is the magnitude of the acceleration?
(b) What is the magnitude of F?
ANSWERS
ANSWERS
(a) 9.8 N
(b) 19.8 m/s
(a) 8.77 m/s2
(b) 26.3 N
13
A man swings his child in a circle of radius 0.75 m. If the
mass of the child is 25 kg and the child makes one
revolution in 1.5 s, what is the magnitude and direction of
the force that must be exerted by the man on the child?
15
ANSWERS
F = 329 N = 73.9 lb directed toward the man.
In the Bohr model of the hydrogen atom, the electron
revolves in a circular orbit around the nucleus. If the radius
is 5.3 x 10-11 m and the electron makes 6.6 x 1015 rev/s,
(a) Find the speed of the electron.
(b) Find the acceleration (magnitude and direction) of the
electron
(c) Find the centripetal force acting on the electron. (This
force is due to the attraction between the positively charged
nucleus and the negatively charged electron.) The mass of
the electron is 9.1 x 10-31 kg.
ANSWERS
(a) 2.2 x 106 m/s
(b) 9.11 x 1022 m/s2 toward the center
(c) 8.29 x 10-8 N
16
A runner moving at a speed of 8.8 m/s rounds a bend with a
radius of 25 m.
(a) Find the centripetal acceleration of the runner.
(b) What supplies the force needed to give this acceleration
to the runner?
ANSWERS
(a) 3.1 m/s2
(b) the track
Chapter 9: Periodic Motion
Assignment
Centripetal Force (a)
17
How much centripetal force is needed to keep a 4 kg iron
ball moving in a horizontal circle of radius 5 m at a velocity
of 15 m/sec?
23
ANSWERS
180 N
18
A 5000 kg airplane makes a horizontal turn l km (1000 m)
in radius at a velocity of 50 m/sec. How much centripetal
force is required?
According to the Guinness Book of World Records, (1990
edition, p. 169) the highest rotary speed ever attained was
2010 m/s (4500 mph). The rotating rod was 15.3 cm (6 in)
long. Assume the speed quoted is that of the end of the rod.
(a) What is the centripetal acceleration of the end of the
rod?
(b) If you were to attach a 1.00-g object to the end of the
rod, what force would be needed to hold it on the rod?
(c) What is the period of rotation of the rod?
ANSWERS
(a) 2.64 x 107 m/s2
(b) 2.64 x 104 N
(c) 4.78 x 10-4 s
ANSWERS
1.25 x 104 N
24
19
Thanks to Earth's rotation, you move with uniform circular
motion. What supplies the force that keeps you moving in
this circle?
ANSWERS
An early major objection to the idea that Earth is spinning
on its axis was that Earth would turn so fast at the equator
that people would be thrown off into space.
(a) Calculate the speed of a 97-kg person at the equator.
The radius of Earth is about 6400 km.
(b) Calculate the centripetal force on the person.
(c) Show the error in this logic by calculating the weight of
the person at the equator.
Earth's Gravity
ANSWERS
20
An athlete whirls a 7.00-kg hammer tied to the end of a 1.3m chain in a horizontal circle. The hammer moves at the
rate of 1.0 rev/s.
(a) What is the centripetal acceleration of the hammer?
(b) What is the tension in the chain?
(a) 465 m/s
(b) 3.3 N
(c) 9.5 x 102 N
25
ANSWERS
(a) 51.2 m/s2
(b) 359.3 N
21
Sue whirls a yo-yo in a horizontal circle. The yo-yo has a
mass of 0.20 kg and is attached to a string 0.80 m long.
(a) If the yo-yo makes 1.0 complete revolution each
second, what force does the string exert on it?
(b) If Sue increases the speed of the yo-yo to 2.0
revolutions per second, what force does the string now
exert?
(c) What is the ratio of answer b to a? Why?
ANSWERS
(a) 6.3 N
(b) 25 N
(c) 4:1
A ball with a mass of 1.2 x 10-2 kg is swung in a horizontal
circle at the end of a string. The string is 1.5 m long and
the ball revolves at a speed of 3.3 m/sec.
(a) What force does the string exert on the ball?
(b) What force does the ball exert on the string?
ANSWERS
(a) .08712 N
(b) .08712 N
26
How fast (in rpm's) must a centrifuge rotate if a particle
10.0 cm from the axis of rotation experiences an
acceleration of 100,000 g's? (remember 1 g is the
acceleration of gravity 9.8 m/s2)
ANSWERS
30,290 rev/min
22
What relationship must exist between an applied force and
the velocity of a moving object if uniform circular motion is
to result?
ANSWERS
Circular motion results when the direction of the force
is constantly perpendicular to the instantaneous
velocity of the object.
Chapter 9: Periodic Motion
Assignment
Centripetal Force (a)
27
In a cyclotron one type of particle the accelerator uses
deuteron (an atom of atomic mass 2 u) which reaches a final
velocity of 10% the speed of light while moving in a
circular path of radius 0.48 m. The deuteron is maintained
in a circular path by a magnetic force. What magnitude of
force is required? Mass of Deuteron = 3.32 x 10-27 kg. The
speed of light is a constant that can be found in your text.
31
A man sits in the seat at the end of a rotating arm 4.8 m
long. What must be the period of revolution of the arm to
cause the man to experience a force 12 times his weight?
ANSWERS
.26 sec/rev
ANSWERS
F = 6.225 x 10-12 N
28
32
Sue whirls a yo-yo horizontally above her head. What is the
direction of the net force that acts on the yo-yo?
ANSWERS
What centripetal force is required to keep a 2-kg mass
moving in a horizontal circle of radius 0.4 m at a speed of 3
m/s?
Along the string toward the center of the circle that
the yo-yo follows.
ANSWERS
45 N
29
33
What is the direction of the force that acts on clothes in the
spin cycle of a washing machine?
ANSWERS
A 3 kg mass attached to a light string rotates in a circular
motion on a horizontal, frictionless table. The radius of the
circle is 0.8 m and the string can support as mass of 25 kg
before breaking. What range of speed can the mass have
before the string breaks?
The walls of the tub push the clothes toward the center
of the tub. Some of the water in the clothes is not
pushed toward the center of the tub, and goes out
through the holes in the wall of the tub. Check out the
pattern in the clothes next to the wall right after the
spin cycle stops.
34
Identify the force that maintains the circular motion of a
rubber stopper swung in a horizontal circle on its chain
ANSWERS
the force in the chain
ANSWERS
8.08 m/s
30
In the Bohr model of the hydrogen atom the velocity of the
electron is approximately 2.2 x 106 m/sec. ( Mass of an
electron = 9.109 x 10-31 kg).
(a) Find the centripetal force acting on the electron as it
revolves in a circular orbit of radius 0.53 x 10-10 m,
(b) Find the centripetal acceleration of the electron.
(c) Find the number of revolutions per second made by the
electron.
ANSWERS
(a) 8.31 x 10-8 N
(b) 9.132 x 1022 m/s2
(c) 6.61 x 1015 rev/sec
35
Imagine that you attach a heavy object to one end of a
spring and then, while holding the spring's other end, whirl
the spring and object in a horizontal circle. Does the spring
stretch? Why? Discuss your answer in terms of the force
that maintains circular motion.
ANSWERS
Yes, the object moves in a straight, inertial path until
the spring force
(F = -kx) is great enough to keep the object at a
constant radius.
Chapter 9: Periodic Motion
Assignment
Centripetal Force (a)
36
A 0.013 kg rubber stopper is attached to a 0.93 m length of
string. The stopper is swung in a horizontal circle, making
one revolution in 1.8 seconds.
(a) Find the speed of the stopper.
(b) Find the centripetal acceleration
(c) Find the force the string exerts on it.
(d) Suppose the mass of the rubber stopper is doubled, but
all the other given quantities remain the same, how would
the velocity, acceleration and force change? Give numerical
answers
(e) If the radius were twice as large , but all other given
quantities remained the same, how would velocity,
acceleration, and force change? Give numerical answers
(f) If the stopper were swung in the same circle so it had a
period half as long, how would the answers change? Give
numerical answers.
39
Identify the force that maintains the circular motion of a
bobsled turning a corner on its track
ANSWERS
the normal force from the curved side of the track
40
A racing car rounds a curve that is banked.
(a) Sketch the auto tire on the incline, drawing vectors
representing all the forces on the tire.
(b) Components of what two forces provide the centripetal
acceleration for the auto tire, and therefore, the auto?
ANSWERS
ANSWERS
(a)
(b)
(c)
(d)
3.25 m/s
11.32 m/s2
0.14 N
3.25.0 m/s
11.32 m/s2
.294 N
(e) 6.48 m/s
22.5 m/s2
.295 N
(f) 6.48m/s
45.28 m/s2
.558 N
37
In a “Rotor-ride” at a carnival, people are rotated in a
cylindrically walled “room.” The room radius is 4.6 m, and
the rotation frequency is 0.50 revolutions per second when
the floor drops out.
(a) What is the minimum coefficient of static friction so
that the people will not slip down?
(b) People on this ride say they were “pressed against the
wall.” Is there really an outward force pressing them against
the wall? If so, what is its source? If not, what is the proper
description of their situation (besides “scary”)? [Hint: First
draw the free-body diagram for a person.]
41
A banked circular curve of highway is designed for traffic
moving at 60 km/h. The radius of the curve is 200 m.
Traffic is moving along the highway at 40 km/h on a stormy
day. What is the minimum coefficient of friction between
tires and road that will allow cars to negotiate the turn
without sliding off the road?
ANSWERS
0.08
42
A curve of radius 30 m is banked so that a car traveling 40
km/h can round it even if the road is so icy that the
coefficient of static friction is approximately zero. Find the
range of speeds at which a car can travel around this curve
without skidding if the coefficient of static friction between
the road and the tires is 0.3.
ANSWERS
Vmax = 56.0 kmlh,
Vmin = 20.1 km/h
ANSWERS
0.22
38
It takes a 615-kg racing car 14.3 s to travel at a uniform
speed around a circular racetrack of 50.0 m radius.
(a) What is the acceleration of the car?
(b) What average force must the track exert on the tires to
produce this acceleration?
43
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock tied to point A by
a string. The rock is moving in a horizontal circle at
constant speed and not resting on a solid surface.
ANSWERS
(a) 9.65 m/s2
(b) 5.94 x 103
ANSWERS
Chapter 9: Periodic Motion
Assignment
Centripetal Force & Hanging Signs (d)
44
Draw the free body diagram of the forces on the rotating
ball.
48
A platform scale calibrated in newtons is placed on the bed
of a truck driven at a constant speed of 14 m/s. A box
weighing 500 N is placed on the scale.
(a) Find the reading on the scale if the truck passes over the
crest of a hill with a radius of curvature 100 m, and
(b) Find the reading on the scale if the truck passes through
the bottom of a dip with a radius of curvature of 80 m.
ANSWERS
(a) 400 N
(b) 625 N
ANSWERS
45
49
Tarzan (m = 85 kg) tries to cross a river by swinging from a
10.0 m long vine. His speed at the bottom of the swing, just
as he clears the water, is 8.0 m/s. Tarzan doesn't know that
the vine has a breaking strength of 1.0 x 103 N. Does he
make it safely across the river? justify your answer.
Why does the water remain in a pail that is whirled in a
vertical path, as shown in the figure below?
ANSWERS
no, 1.37 x 103 N > 1.0 x 103 N
50
A 515 kg roller-coaster car rolls down past point A and then
up past point B, as shown in the figure.
(a) If the vehicle has a speed of 20.0 m/s at point A, what is
the force of the track on the vehicle at this point?
(b) What is the maximum speed the vehicle can have at B
for gravity to hold it on the track?
ANSWERS
water tends to follow a straight-line path; pail exerts
force that maintains circular motion
46
A certain string can withstand a maximum tension of 9.0 N.
without breaking. A child ties a 0.82 N. stone to one end.
Holding the other end, he whirls the stone in a vertical
circle of radius 3.0 m slowly increasing the speed until the
string breaks.
(a) Where is the stone on its path when the string breaks?
(b) What is the speed of the stone as the string breaks?
ANSWERS
(a) 25,647 N
(b) 12.1 m/s
ANSWERS
(a) At the bottom of circle
(b) 17.1 m/s
47
A man swings a pail of water in a vertical circle 3.1 meters
in radius.
(a) If the water is not to spill, what is the minimum velocity
the pail can have.
(b) How much time per revolution is this equivalent to?
ANSWERS
(a) 5.51 m/s
(b) 3.53 sec/rev
51
With an average mass of only 30.0 g, the mouse lemur of
Madagascar is the smallest primate on Earth. Suppose this
lemur is swinging on a light vine with a length of 2.4 m so
that the lemur's tangential speed at the bottom point is 2.8
m/s. Calculate the tension in the vine at that point.
ANSWERS
0.393 N
Chapter 9: Periodic Motion
Assignment
Net Forces and Vertical Circles (e)
52
In 1992, a team of 12 athletes from Great Britain and
Canada rappelled 446 m down the CN Tower in Toronto,
Canada. Suppose an athlete with a mass of 75 kg, having
reached the ground, takes a joyful swing on the 446 m long
rope.
(a) If the speed of the athlete at the bottom point of the
swing is 12 m/s, what is the force that maintains the
athlete's circular motion?
(b) What is the tension in the rope? Neglect the rope's mass.
56
ANSWERS
3.13 m/s
ANSWERS
57
24 N
760 N
53
A pail of water is rotated in a vertical circle of radius 1 m
(the approximate length of a person's arm). What is the
minimum speed of the pail at the top of the circle if no
water is to spill out?
A stunt man drives a car over the top of a hill, the cross
section of which can be approximated by a circle of radius
250 m, as in the figure. What is the greatest speed at which
he can drive without the car leaving the road at the top of
the hill?
A ball attached to the end of a string 0.8 m in length is
rotated in a vertical circle. Determine the minimum speed
of the ball at the top of its path if it maintains a circular
path. (Note that below this speed, the tension in the string
is zero at the top).
ANSWERS
2.8 m/s
58
A jet plane traveling 1800 km/hr (500 m/s) pulls out of a
dive by moving in an arc of radius 3.0 km. What is the
plane's acceleration in g's.
ANSWERS
ANSWERS
93.1 m/s = 9.5 g's
178 km/h
49.497 m/s
59
54
A 150 N. student on a steadily rotating Ferris wheel has an
apparent weight (net force) of 125 N. at the highest point.
(a) What is his apparent weight (net force) at the lowest
point?
(b) What would be his apparent weight (net force) at the
highest point if the speed of the Ferris wheel were doubled?
ANSWERS
8.4 m/s (no gravity effect)
5.6 m/s (with gravity effect)
ANSWERS
(a)
(b)
55
Tarzan plans to cross a gorge by swinging in an arc from a
hanging vine. If his arms are capable of exerting a force of
1500 N on the rope. What is the maximum speed he can
tolerate at the lowest point of his swing? His mass is 85 kg;
the vine is 4.0 meters long.
60
A ball on the end of a string is revolving at a uniform rate in
a vertical circle of radius 1.10 m. Its speed is 3.75 m/s and
its mass is 0.355 kg.
(a) Calculate the tension at the top of its path.
(b) Calculate the tension at the bottom of its path?
A block of mass m at the end of a string is whirled around
in a vertical circle of radius R. Find the critical speed
below which the string would become slack at the highest
point.
ANSWERS
ANSWERS
(a) 1.06 N
(b) 8.017 N
(gR)1/2
61
What minimum speed must a roller coaster be traveling
when upside down at the top of a circle if the passengers are
not to fall out? Assume a radius of curvature of 8.0 m.
ANSWERS
v = 8.854 m/s
Chapter 9: Periodic Motion
Assignment
Net Forces and Vertical Circles (e)
62
A carnival clown rides a motorcycle down a ramp and
around a "loop-the-loop." If the loop has a radius of 18 m,
what is the slowest speed the rider can have at the top of the
loop to avoid falling? Hint: At this slowest speed, at the top
of the loop, the clown's weight is equal to the centripetal
force.
67
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock swinging on a
rope.
ANSWERS
10 m/s
63
A 75-kg pilot flies a plane in a loop. At the top of the loop,
where the plane is completely upside-down for an instant,
the pilot hangs freely in the seat and does not push against
the seat belt. The airspeed indicator reads 120 m/s. What is
the radius of the plane's loop?
ANSWERS
ANSWERS
3
1.5 x 10 m
64
A 2.0-kg object is attached to a 1.5 m long string and swung
in a vertical circle at a constant speed of 12 m/s.
(a) What is the tension in the string when the object is at
the bottom of its path?
(b) What is the tension in the string when the object is at
the top of its path?
68
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock tied to point B and
moving downward in a vertical circle with string horizontal.
ANSWERS
(a) 2.1 x 102 N
(b) 1.7 x 102 N
65
A roller coaster moves through a vertical loop at a constant
speed, suspending its passengers upside down. In what
direction is the force that causes the coaster and its
passengers to move in a circle? What provides this force?
ANSWERS
ANSWERS
toward the center of the track;
the track
66
A girl at a state fair swings a ball in a vertical circle at the
end of a string. Is the force applied by the string greater
than the weight of the ball at the bottom of the ball's path?
ANSWERS
Yes, the string must exert a force equal to the ball's
weight and the circular force that maintains circular
motion.
69
When you drive rapidly on a hilly road or ride in a roller
coaster you feel lighter as you go over the top of a hill, and
heavier when you go through a valley. Sketch the situation,
showing the forces that explain this sensation.
ANSWERS
To move you in a circular path, there is an added
upward force on you. Since you react to external
forces, t his makes you feel heavier or lighter
Chapter 9: Periodic Motion
Assignment
Net Forces and Vertical Circles (e)
70
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock swinging on a
rope, at the top of a vertical circle.
74
A mass m = 5 kg on a frictionless table is attached to a
hanging mass M = 8 kg by a cord of radius 0.5 m through a
hole in the table shown in the figure. Find the speed with
which m must spin for M to stay at rest.
ANSWERS
ANSWERS
71
A string 1 meter long is used to whirl a ball of unknown
mass in a vertical circle. What is the minimum velocity of
the ball if the string is to be just taunt when the ball is at the
top of the circle?
75
ANSWERS
3.13 m/s
72
A girl is whirling a ball on a string around her head in a
horizontal plane. She wants to let go at precisely the right
time so that the ball will hit a target on the other side of the
yard. When should she let go of the string?
ANSWERS
(a) toward the Center
(b) 0.61 m/s2
1.2 m/s2
1.8 m/s2
(c) friction
(d) 15 cm
ANSWERS
73
A coin is placed on a stereo record revolving at 33 1/3
revolutions per minute.
(a) In what direction is the acceleration of the coin, if any?
(b) Find the acceleration of the coin when it is placed 5.0,
10, and 15 cm from the center of the record.
(c) What force accelerates the coin?
(d) At which of the three radii listed in b would the coin be
most likely to fly off? Why?
A mass m = .4 kg on a frictionless table is attached to a
hanging mass M = 10 kg by a cord of radius 2.0 m through
a hole in the table shown in the figure. Find the speed with
which m must spin for M to stay at rest.
76
Friction provides the centripetal force necessary for a car to
travel around a flat circular race track. What is the
maximum speed at which a car can safely travel around a
circular track of radius 80.0 m if the coefficient of friction
between the tire and road is 0.30?
ANSWERS
15 m/s
77
ANSWERS
22,2 n.s
A 60.0-kg speed skater with a velocity of 18.0 m/s comes
into a curve of 20.0-m radius. How much friction must be
exerted between the skates and ice to negotiate the curve?
ANSWERS
972 N
Chapter 9: Periodic Motion
Assignment
Friction and Centripetal Forces (b)
78
A 2.00 x 103 kg car rounds a circular turn of radius 20.0 m.
If the road is flat and the coefficient of static friction
between the tires and the road is 0.55, how fast can the car
go without skidding?
83
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock riding on a
horizontal disk that is rotating at constant speed about Its
vertical axis. Friction prevents rock from sliding.
ANSWERS
10.38 m/s
79
A 13,500 N car traveling at 50.0 km/h rounds a curve of
radius 2.00 x 102 m.
(a) Find the centripetal acceleration of the car
(b) Find the force that maintains centripetal acceleration
(c) Find the minimum coefficient of static friction between
the tires and the road that will allow the car to round the
curve safely
ANSWERS
ANSWERS
84
(a) 0.965 m/s2
(b) 1.33 x 103 N
(c) 0.0985
80
If the coefficient of static friction for tires on a road is 0.25,
at what greatest speed can a car round a level 150-ft (= 47.5
m) radius curve without slipping?
A 29.4 kg girl sits in a tire that is attached to an
overhanging tree limb by a rope. The girl's father pushes
her with a linear speed of 2.5 m/s so that she travels in a
horizontal circle with a diameter of 4.2 m. What is the
magnitude of the force that maintains the girl's circular
motion?
ANSWERS
88 N
ANSWERS
34.6 ft/s
(10.8 m/s)
81
A child places a picnic basket on the outer rim of a merrygo-round that has a radius of 4.6 m and revolves once every
30 s.
(a) What is the speed of a point on the rim of the merry-goround?
(b) How large must the coefficient of static friction be for
the basket to stay on the merry-go-round?
85
A test car moves around a 3.25 km circular track at a linear
speed of 35.0 m/s. Find the magnitude of the force that
maintains the car's circular motion if the test car has a mass
of 905 kg.
ANSWERS
2.14 x 103 N
86
Identify the force that maintains the circular motion of a
bicycle moving around a flat circular track
ANSWERS
ANSWERS
(a) .96 m/s
(b) 0.0206
82
A 100-g disk sits on a horizontally rotating turntable. The
turntable makes one revolution each second. The disk is
located 10 cm from the axis of rotation of the turntable.
(a) What is the frictional force acting on the disk?
(b) The disk will slide off the turntable if it is located at a
radius larger than 16 cm from the axis of rotation. What is
the coefficient of static friction?
friction between the tires and the track
87
A 25 kg child moves with a speed of 1.80 m/s when he is
12.4 m from the center of a merry-go-round.
(a) Calculate the centripetal acceleration of the child
(b) Calculate the net force exerted on the child.
ANSWERS
ANSWERS
(a) 0.395 N
(b) 0.644
(a) .26 m/s2
(b) 6.53 N
Chapter 9: Periodic Motion
Assignment
Friction and Centripetal Forces (b)
88
A motorcycle begins to skid when it makes a turn of 50
meter radius at a velocity of 40 m/sec. What is the highest
velocity at which it can make a turn of 100 meter radius?
92
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock stuck by friction
against the Inside wall of a drum rotating about Its vertical
axis at constant speed.
ANSWERS
56.57 m/s
89
Racing on a flat track, a car going 32 m/s rounds a curve 56
m in radius.
(a) What is the car's centripetal acceleration?
(b) What would be the minimum coefficient of static
friction between tires and road that would be needed for the
car to round the curve without skidding?
ANSWERS
ANSWERS
(a) 18 m/s2
(b) 1.8
93
90
In an amusement-park ride, riders stand against the wall of
a spinning cylinder. The floor falls away and the riders are
held up by friction. If the radius of the cylinder is 4 m, find
the minimum number of revolutions per minute necessary
when the coefficient of friction between a rider and the wall
is 0.4.
A 50 kg skier starting from rest coasts down a frictionless
30 degree hill 40 m long enters a frictionless horizontal
stretch then turns sharply in a horizontal circle of radius 30
m. What is the horizontal force exerted (by what) on the
skier during the turn.
ANSWERS
653 N
ANSWERS
23.6 rev/min
94
91
Show with a force diagram how a motorcycle can travel in
a circle on the inside vertical wall of a hollow cylinder.
Assume reasonable parameters (coefficient of friction,
radius of the circle, mass of the motorcycle, or whatever is
required), and calculate the minimum speed needed.
A boy whirls a stone in a horizontal circle 2.0 m above the
ground by means of a rope 1.5 meters long. The string
breaks and the stone flies off horizontally striking the
ground 10 meters away from the release point.
(a) How long was the stone in the air after it was released?
(b) What was the centripetal acceleration of the stone while
it was in circular motion.
ANSWERS
ANSWERS
Vmin = 14.0 m/s
= 31.3 mi/h
(a) .64 sec
(b) 162.7 m/s2
15.65 m/s
Chapter 9: Periodic Motion
Assignment
Mixed System Problems (k)
95
A 1.34 kg ball is attached to a rigid vertical rod by means of
two massless strings each 1.7 m apart. The strings are
attached to the rod at points 1.7 m apart. The system is
rotating about the axis of the rod, both strings being taut
and forming an equilateral triangle with the rod as shown in
the figure below The tension in the upper string is 35 N
(a) Draw the free-body diagram for the ball.
(b) What is the tension in the lower string?
(c) What is the net force on the ball at the instant shown in
the figure?
(d) What is the speed of the ball?
97
The rock shown below is acted on by one or more forces.
Draw the free body diagram for the rock resting against the
frictionless inside wall of a cone rotating about its vertical
axis at constant speed. Not accelerating vertically.
ANSWERS
98
ANSWERS
(a)
(b) 8.74 N
(c) 37.9 N
(d) 6.45 m/s
96
In a popular amusement-park ride, a cylinder of radius 3.00
m is set in rotation at an angular speed of 5.00 rad/s, as
shown in the figure below. The floor then drops away,
leaving the riders suspended against the wall in a vertical
position. What minimum coefficient of friction between a
rider's clothing and the wall of the cylinder is needed to
keep the rider from slipping?
The figure below shows the path taken by an automobile. It
consists of straight lines and arcs of circles. The automobile
starts from rest at point A and accelerates until it reaches
point B. It then proceeds at constant speed until it reaches
point E. From point E on it slows down, coming to rest at
point F. What is the direction of the net force, if there is
any, on the automobile at the midpoint of each section of
the path?
ANSWERS
0.131
99
ANSWERS
AB, force vertically upward;
BC, force radial toward center of circular arc;
CD, no force;
DE, force is toward center of circular arc;
EF, force acts vertically upward.
A piece cut from a bicycle inner tube is stretched to a total
length of 0.7 m long when it carries a load of 25 N. The
force constant k is 300 N/m.
(a) How far did the rubber tube stretch when 75 Newtons
of force were added?
(b) What was the original length of the tube when no load
was present. (Hint: You need to find the change in x (draw
all three pictures)
Answer
(a) .25
(b) .616 m
Chapter 9: Periodic Motion
Assignment
Hooke's Law (f)
100 Answer the following questions.
(a) What are the dimensions of Hooke's constant
(b) What are the dimensions of the spring constant
(c) What are the dimensions of the force constant?
105 Answer the following questions.
(a) What are the dimensions of period ?
(b) What are the dimensions of frequency?
Answer
Answer
(a) N/cm
(b) N/cm
(c) N/cm
sec/cyc
cyc/sec
106 A vibrating reed in a harmonica makes 200 vib/sec. What
is the period?
101 A spring is 6 cm long. When it is suspended vertically and
a 200 g mass is hung from its lower end, its length increases
to 7.5 cm.
(a) What is the constant of the spring?
(b) What is the tension in the spring if it is stretched to 8.7
cm?
Answer
(a) 1.34 N/cm
102 A spring which obeys Hooke's law is stretched 4 cm by a
force of 6 N.
(a) What is the value of k, the spring constant?
(b) How much will the spring be stretched by a force of 1.5
N?
Answer
.005 sec
107 What is the time required for 2400 vibrations of a loaded
spring if the frequency is 4 vib/sec?
Answer
600 sec.
108 If the force constant of a spring is 8 N/m, what suspended
mass will give a period of 1 sec?
Answer
0.2 kg
Answer
(a) 1.5 N/cm
(b) 1 cm
103 A 250 g block is placed on top of a vertical spring with a
spring constant k = 2.5 N/cm. How much is the spring
compressed?
Answer
.980 c,
104 A block is placed on a spring whose spring constant is 2.0
N/cm and compresses the spring 10 cm.
(a) What is the weight of the block?
(b) What is the mass of the block?
109 The piston of an engine moves a total distance of 0.2 m
from one extreme point to the other.
(a) Draw the piston at the top, middle and bottom of its
cycle.
(b) What is the amplitude?
(c) What is the displacement when the piston is 0.1 m from
one end of its stroke?
Answer
(a)
(b) 0.1 m
(c) 0.0
110 The piston in problem 5 makes 2 vib/sec. How fast is it
moving when passing through the midpoint of its path?
Answer
Answer
(a) 20 N
(b) 2.04 kg
1.26 m/sec
111 Compute the time for one vibration of a spring whose force
constant is 16 N/m, if the load has a mass of l kg.
Answer
1.57 sec
Chapter 9: Periodic Motion
Assignment
Springs (i)
112 A point on a horizontal vibrating piano string is moving up
and down in SHM with a frequency of 200 vib/sec. The
maximum displacement of the point below its normal
position is 0.5 mm.
(a) What is the maximum velocity of the particle, and
where is this velocity reached?
(b) What is the maximum acceleration of the particle?
117 A small electronic computer placed on board a satellite
must withstand a maximum acceleration of 10 G's. To test
this, the computer is attached to a horizontal table which is
driven from side to side in SHM at 20 vib/s. What must be
the amplitude of vibration of this shake-test apparatus in
order to give a maximum acceleration of 10 G's?
Answer
Answer
(a) 0.628 m/sec
(b) 789 m/sec2
113 A 30 g load hangs from a long, light spring. When pulled
down 20 cm below its equilibrium position and released, it
vibrates with a period of 2 sec.
(a) What is its velocity as it passes through the equilibrium
position?
(b) By how much will the spring shorten if the load is
removed?
Answer
(a) 62.8 cm/sec
(b) 0.99 m
114 With what acceleration must be given to a 20 kg mass
suspended from a spring so that it will oscillate in SHM
with a period of 0.5 sec and an amplitude of 2 cm.
Answer
3.15 m/sec2
115 A raft of weight 4000 N is floating in a pond. When a 1000
N man climbs on board, the raft sinks 0.04 m deeper into
the water to a new equilibrium position. If the man rolls
off, how many vertical vibrations does the empty raft make
in 5 seconds?
Answer
6.25 vib
when the mans rolls off boat
116 A light rubber band is hanging loosely from the ceiling. A
load is attached to the free end of the rubber band, and it is
observed that the load, when released, moves downward a
distance of 40 cm before starting to rise again. What is the
period of this SHM? (Hint: The mass is not given. Assign
to it the value of m, on the chance that it will cancel out of
your final equation for the period?
Answer
0.898 sec
.0062 m
118 Describe a method of using a simple pendulum to find the
value of g at a given location.
Answer
Use the equation 2π√(l/g) or
g = (4π2l)/T2
measure the length of the pendulum and its period,
and calculate g.
119 What is the length of a simple pendulum whose period is
l.00 s? Assume normal g.
Answer
0.248 m
120 Compute the length of a clock pendulum that ticks once
each second. (Hint: It ticks twice during each complete
vibration.)
Answer
.999 m
121 What would be the period of a pendulum suspended from
the top of a tall building with a light string 353 m long?
Answer
37.7 sec.
122 The value of the acceleration due to gravity on the moon is
about 1/6 that on the earth. What would be the period on
the moon of a simple pendulum whose period on the earth
is 2.0 sec?
Answer
4.90 sec
123 Sue is swinging her yo-yo as if it was a pendulum. At what
point in the swing does the yo-yo have the greatest force
acting on it?
Answer
At the bottom of the swing
Chapter 9: Periodic Motion
Assignment
Pendulums (j)
124 A future astronaut lands on a planet with an unknown value
of g. She finds that the period of a pendulum 0.65 m long is
2.8 s. What is g for the surface of this planet?
Answer
8
131 If the centers of Earth and the moon are 3.9 x 10 m apart,
the gravitational force between them is about 1.9 x 1020 N.
What is the approximate mass of the moon?
7.2 x 1022 kg
3.3 m/s2
125 A pendulum has a length of 0.67 m.
(a) Find its period.
(b) How long would the pendulum have to be to double the
period?
(c) Why is your answer to part b not just double the length?
132 Using Newton's Universal Law of Gravity
(a) What is the gravitational force between two spherical
8.00 kg masses that are 5.0 m apart?
(b) What is the gravitational force between them when they
are 5.0 x 101 m apart?
(a) 1.7 x 10-10 N
(b) 1.7 x 10-12 N
Answer
force between two electrons 1.00 m apart
133 The gravitational
is 5.42 x l0-71 N. Find the mass of an electron.
(a) 1.6 s
(b) 2.7 m
(c) T=√l
9.01 x 10-31 kg
126 For each of the examples of two-body gravitational action
below, find the missing quantity for the data given. (G =
6.7 x 10-11 (Nm2/kg2)
a
b
c
d
(a)
(b)
(c)
(d)
Force
N
A
0.13
1.00
6.00
Mass 1
kg
1015
B
1030
1018
Mass 2
kg
1014
1/0
C
1013
Dist
m
106
104
1010
D
6.7 x 106 N
1.94 x 1017 kg
1.5 kg
1.06 x 1010 m
134 Compute the gravitational force the sun exerts on Jupiter.
4.17 x 1023 N
8
135 Two ships each of 10 kg mass are moored near each other.
Assume the effective distance between them as far as
gravitational attraction is concerned is 100 m. What
gravitational force do they exert on one another?
671 N
136 What is the force of attraction between two spherical 100
kg masses whose centers are 2.00 m apart?
1.67 x 10-7 N
127 Two satellites are in circular orbits about Earth, one 150
km above the surface, the other 160 km.
(a) Which satellite has the larger orbital period?
(b) Which one has the larger velocity?
(a) The one at 160 km has the larger period.
(b) The one at 150 km has the larger velocity.
is 9.1 x 10-3l kg. The mass of a
128 The mass of an electron
proton is 1.7 x 10-27 kg. They are about 1.0 x 10-10 m apart in
a hydrogen atom. What gravitational force exists between
the proton and the electron of a hydrogen atom?
1.0 x 10-47 N
129 Two 1.00-kg masses have their centers 1.00 m apart. What
is the force of attraction between them?
6.67 x 10-11 N
Their
130 Two large spheres are suspended close to each other.
centers are 4.0 m apart. One sphere weighs 9.8 x 102 N. The
other sphere has a weight of 1.96 x 102 N. What is the
gravitational force between them?
8.3 x 10-9 N
137 From the data on the planetary data sheet, calculate the
gravitational force on the earth due to the sun. It is this
force which holds the earth in its orbit.
3.6 x 1022 N
is placed in a circular orbit with a radius of 1.0 x
138 A satellite
107 m a period of 9.9 x 103 s. Calculate the mass of Earth.
Hint: Gravity supplies the needed centripetal force for such
a satellite. Scientists have actually measured the mass of
Earth this way.
6.0 x 1024 kg
139 Find the speed and period of a satellite that would orbit
Mars 175 km above its surface.
V = 3.47 x 103 m/s
T = 6.45 x 103 s
or
1.79 h
Chapter 17: Universal Gravity
Assignment
Universal Gravity and Centripetal Force (e)
140 The following problem examins some characteristics of the
planet Mercury.
(a) Find the speed of a satellite in orbit 265 km above
Mercury’s surface.
(b) Find the the period of the satellite orbit 265 km above
Mercury’s surface.
(a) 2.96 x 103 m/s
(b) 85.9 min
141 Using Astronomical data.
(a) Find the velocity with which Mercury around the sun
(b) Also, find the velocity of Saturn.
(c) Now , comment on whether or not it mae sense that
Mercury is named after a speedy messenger of the gods,
while Saturn is named after the father or Jupiter.
(a) 4.79 x 104 m/s
(b) 9.65 x 103 m/s
(c) about as 1/5 as fast as Mercury
142 Using Astronomical Data
(a) Calculate the velocity that a satellite shot from
Newton's cannon must have in order to orbit Earth, 150 km
above its surface.
(b) How long would it take for the satellite to return to the
cannon in seconds and minutes?
(a) 7.8 x 103 m/s
(b) 84 min 18 sec
a moon of Saturn, has an orbital radius of 1.87 x
143 Mimas,
108 m and an orbital period of about 23 h. Use Newton's
version of Kepler's third law and these data to find the mass
of Saturn.
5.6 x 1025 kg
146 A 10, 000 kg spaceship is drifting on a long mission toward
the outer edge of the solar system. It has put out a small
experimental satellite which revolves around it at a distance
of 120 meters under their mutual gravitational attraction.
(a) What is the period of revolution of the satellite?
(b) What is the speed of the satellite?
(a) 1.01 x 107 s
(b) 0.075 m/s or
7.5 x 10-5 m/s
147 Assume the earth is perfectly round and has a radius of
6400 km.
(a) What is the weight of a 100 kg man at the North Pole?
(b) What is the centripetal force of a 100 kg man at the
Equator?
(c) How much less does a man with a mass of 100 kg
apparently weigh at the equator than at the poles because of
the rotation of the earth?
(d) How fast would the earth have to spin in order that he
would exert no force on a scale at the equator?
(e) How many times larger is the speed of rotation in d
than the actual speed?
(a)
(b)
(c)
(d)
980 N
3.37 N
976.6 N
5065.6 S or
1.4 hours, or
7.935 m/s
(e) 17 times greater
148 If you weigh 637 N on Earth's surface, how much would
you weigh on the planet Mars? (Mars has a mass of 6.37 x
1023 kg and a radius of 3.43 x 105 m.)
235 N
144 A geosynchronous satellite appears to remain over one spot
on Earth. A geosynchronous satellite has an orbital radius
of 4.23 x 107 m.
(a) Calculate its speed in orbit.
(b) Calculate its period.
(a) 3.07 x 103 m/s
(b) 24.0 h
145 On July 19, 1969, Apollo lI's orbit around the moon was
adjusted to an average orbit of 111 km. The radius of the
moon is 1785 km and the mass of the moon is 7.3 x 1022 kg.
(a) How many minutes did it take to orbit once?
(b) At what velocity did it orbit the moon?
(a) 1.2 x 102 min
(b) 1.6 x 103 m/s
149 What would be the value of g, acceleration of gravity, if
Earth's mass was double its actual value, but its radius
remained the same? If the radius was doubled, but the mass
remained the same? If both the mass and radius were
doubled
19.6 m/s2
2.45 m/s2
4.9 m/s2
150 What would be the strength of Earth's gravitational field at
a point where an 80.0 kg astronaut would experience a 25%
reduction in weight?
7.35 m/s2
151 On the surface of the moon, a 91.0 kg physics teacher
weighs only 145.6 N. What is the value of the moon's
gravitational field at its surface?
1.60 m/s2
Chapter 17: Universal Gravity
Assignment
Universal Gravity and Weight (d)
152 Two satellites of equal mass are put into orbit 30-7m apart.
The gravitational force between them is 2.0 x 10 N.
(a) What is the mass of each satellite?
(b) What is the initial acceleration given to each satellite by
the gravitational force?
(a) 1.6 x 103 kg
(b) 1.3 x 10-10 m/s2
153 A force of 40.0 N is required to pull a 10.0 kg wooden
block at a constant velocity across a smooth glass surface
on Earth. What force would be required to pull the same
wooden block across the same glass surface on the planet
Jupiter?
100 N
20
154 The asteroid Ceres has a mass 7 x 10 kg and a radius of
500 km.
(a) What is g on the surface?
(b) How much would a 85-kg astronaut weigh on Ceres?
(a) 0.2 m/s2
(b) 2 x 101 N
3
3
155 The radius of Earth is about 6.40 x 10 km. A 7.20 x 10 N
spacecraft is traveling away from Earth.
(a) What is the weight of the spacecraft 6.40 x 103 km from
the Earth's surface?
(b) What is the weight of the spacecraft 1.28 x 104 km from
the Earth's surface?
(a) 1.80 x 103 N
(b) 800 N
156 How high does a rocket have to go above Earth's surface
until its weight is half what it would be on Earth?
9.05 x 105 m
157 At what height above the earth's surface will a rocket have
half the force of gravitation on it that it would have at sea
level? Express your answer in earth radii.
158 The instrument-carrying payload of a rocket weighs 1058
newtons on the earth. What does it weigh 2.560 x 104 km
above the earth?
42 N
159 Find the weight of a 100 kg man on Jupiter.
2469 N
for the acceleration due to
160 Calculate the theoretical value
gravity at a point 1.00 x 107 m from the center of the earth.
3.98 m/s2
161 At what height above the earth's surface will a rocket
experience just a quarter the pull from the earth that it feels
at the earth's surface?
1.28 x 107 m
2 Earth radii from the center of the Earth
1 Earth radius above the surface of the earth
22
162 The 5mass and radius of the moon are 7.3 x 10 kg and 1.74
x 10 m respectively.
(a) Calculate the value of g on the moon
(b) Calculate the weight of a 50 kg boy on the earth
(c) Calculate the weight of a 50 kg boy on the moon
(d) Calculate the time it takes of an object to fall 4.9 m on
the moon.
(e) Calculate the time it takes of an object to fall 4.9 m
both on the earth
(f) Many physics students would like to change the
magnitude of g from 9.8 m/s2 to 10 m/s2. To do this how
should the mass and the radius of the earth be changed?
(g) How could the same change in g be accomplished,
theoretically, without any change in the mass or radius of
the earth?
(a)
(b)
(c)
(d)
(e)
(f)
1.6 m/s2
F earth = 490 N
F moon = 80 N
2.4 sec Moon
1 second Earth
Only if G is not constant g.
163 Use the Universal Law of Gravity to determine the
following:
(a) What is the weight of a 1.0 kg mass one earth radius
from the surface of the earth (Radius of the earth = 6.4 x
105 m or 4000 miles).
(b) At what distance from the surface is the weight of any
mass reduced one-half?
(a) 2.5 N
(b) 1.4 Earth radii from the center of the earth
.4 radii from the surface of the earth
164 In a “Rotor-ride” at a carnival, people are rotated in a
cylindrically walled “room.” The room radius is 7.6 m, and
the rotation frequency is 0.25 revolutions per second when
the floor drops out.
(a) What is the minimum coefficient of static friction so
that the people will not slip down?
(b) People on this ride say they were “pressed against the
wall.” Is there really an outward force pressing them against
the wall? If so, what is its source? If not, what is the proper
description of their situation (besides “scary”)? [Hint: First
draw the free-body diagram for a person.]
ANSWERS
Chapter 9: Periodic Motion
Assignment
Mixed System Problems (k)
165 In a “Rotor-ride” at a carnival, people are rotated in a
cylindrically walled “room.” The room radius is 3.6 m, and
the rotation frequency is 1.50 revolutions per second when
the floor drops out.
(a) What is the minimum coefficient of static friction so
that the people will not slip down?
(b) People on this ride say they were “pressed against the
wall.” Is there really an outward force pressing them against
the wall? If so, what is its source? If not, what is the proper
description of their situation (besides “scary”)? [Hint: First
draw the free-body diagram for a person.]
ANSWERS
0.22
166 In a “Rotor-ride” at a carnival, people are rotated in a
cylindrically walled “room.” The room radius is 5.6 m, and
the rotation frequency is 0.50 revolutions per second when
the floor drops out.
(a) What is the minimum coefficient of static friction so
that the people will not slip down?
(b) People on this ride say they were “pressed against the
wall.” Is there really an outward force pressing them against
the wall? If so, what is its source? If not, what is the proper
description of their situation (besides “scary”)? [Hint: First
draw the free-body diagram for a person.]
ANSWERS