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REVIEW
Chapter 3
Understanding Concepts
1. Is centripetal acceleration an instantaneous accelera-
tion, an average acceleration, both, or neither?
Explain.
2. Do all points along the minute hand of a clock expe-
rience the same centripetal acceleration? Explain.
3. If the speed of a particle in circular motion is
increasing, is the net acceleration of the particle still
directed toward the centre of the circle? Use a diagram to explain your answer.
4. A civil engineer has calculated that the magnitude of
the maximum centripetal acceleration of a car on a
certain horizontal curve is 4.4 m/s2. What is the
minimum radius of this curve for a car travelling at
25 m/s?
5. For the clock shown in Figure 1, calculate the magni-
tude of the centripetal acceleration of the tip of the
second hand, the minute hand, and the hour hand.
12
10
1
2
9.8 cm
6.0 cm
9
8.0 cm
6
4
5
Figure 1
6. A chicken is cooking on the rotating turntable of a
microwave oven. The end of the drumstick, which is
16 cm from the centre of rotation, experiences a centripetal acceleration of magnitude 0.22 m/s2.
Determine the period of rotation of the plate.
7. For each of the following situations, draw an FBD
and name the force(s) causing the centripetal
acceleration:
(a) A truck travels, without sliding, around an
unbanked curve on a highway.
(b) A bus travels around a banked curve at the
optimal speed for the banking angle.
(c) A planet travels in an essentially circular orbit
around the Sun.
(d) A communications satellite travels in a circular
orbit around Earth.
NEL
circle of radius 26 cm in the spin cycle of a washing
machine. The frequency of rotation is 4.6 Hz.
(a) Name the force causing the centripetal acceleration. What object exerts that force?
(b) What is the speed of the towel?
(c) Determine the magnitude of the centripetal force
on the towel.
9. Neptune travels in a nearly circular orbit, of diameter
9.0 1012 m, around the Sun. The mass of Neptune
is 1.0 1026 kg. The gravitational force of attraction
between Neptune and the Sun has a magnitude of
6.8 1020 N.
(a) What is the speed of Neptune?
(b) Determine Neptune’s period of revolution
around the Sun in Earth years.
10. Points A through E in Figure 2 represent a piece of
cement experiencing centripetal acceleration in the
vertical plane inside a rotating cement mixer. The
mixer itself is in uniform circular motion. For each of
the points A through E, draw an FBD of the piece of
cement at that point, and state what forces cause the
centripetal acceleration.
E
3
8
7
8. A wet towel, of mass 0.65 kg, travels in a horizontal
on of rotati
ecti
on
dir
11
Unit 1
D
C
B
A
Figure 2
11. A 45.7-kg boy on a swing moves in a circular arc of
radius 3.80 m. At the lowest position, the child’s
speed reaches 2.78 m/s. Determine the magnitude of
the tension in each of the two vertical support chains.
12. A sport utility vehicle, of mass 2.1 103 kg, travels in
the horizontal plane around an unbanked curve of
radius 275 m at a speed of 26 m/s, without sliding.
(a) Determine the minimum coefficient of static
friction between the tires and the road.
(b) How would your answer in (a) be affected if the
mass of the vehicle were greater because of the
presence of heavy cargo?
(c) How would your answer in (a) be affected if the
curve were sharper (i.e., if its radius were
smaller)?
Circular Motion 159
13. A ball, of mass 0.23 kg, is attached securely to a
string, then whirled at a constant speed in a vertical
circle of radius 75 cm.
(a) Draw FBDs of the ball at the top and the bottom
of the circle.
(b) Determine the magnitude of the tension in the
string at the locations in (a) for which the speed
of the ball is 3.6 m/s.
(c) Calculate the minimum speed of the ball at the
top of the path if it is to follow a complete circle.
20. The Canadarm2 is the robotic arm, designed and
built in Canada, that services the ISS in its orbit
4.50 102 km above the surface of Earth. Although
the mass of this arm is 1.80 103 kg, it can move
masses as large as 1.16 105 kg on the ISS.
(a) Determine the magnitude of the force of gravity
acting on the maximum load for the arm.
(b) If the arm had to move such a large mass here on
the surface of Earth, it would break. Why does it
not break in space?
14. In which of the following situations would it not be
possible to determine the gravitational force of
GmAmB
attraction from the equation FG ,
if in
r2
each case the masses and the radius r are provided?
(a) Saturn and one of Saturn’s moons
(b) two friends hugging
(c) a ball moving in a parabola through the air and
Earth
(d) two textbooks standing together on a bookshelf
15. A spherical meteor approaches Earth from a great
distance. By what factor does the force between Earth
and the meteor increase when the distance between
the centres of the two bodies decreases by a factor
of 3.9?
16. At a certain distance above Earth’s surface, the gravi-
tational force on a certain object is only 2.8% of its
value at Earth’s surface. Determine this distance,
expressing it as a multiple of Earth’s radius, rE.
Applying Inquiry Skills
21. Suppose that you have determined the results in
Table 1 while performing an investigation. Determine
the new value for the centripetal force.
Table 1 Data for Question 21
Before
After
mass 1 ball
mass 3 balls
radius 0.75 m
radius 1.50 m
frequency 1.5 Hz
frequency 3.0 Hz
centripetal force 8.0 units centripetal force ? units
22. A conical pendulum consists of a mass (the pendulum
bob) that travels in a circle on the end of a string,
tracing out a cone as in Figure 4. For the pendulum
shown, m 1.50 kg, L 1.15 m, and v 27.5°.
17. Determine the magnitude of the gravitational force
between two bowling balls, each of mass 1.62 kg, if
the centres are separated by 64.5 cm.
v
L
18. The orbit of Venus is approximately circular. The
masses of the Sun and Venus are 1.99 1030 kg and
4.83 1024 kg, respectively. The Sun-Venus distance
is 1.08 108 km. Determine the centripetal acceleration of Venus.
r
19. Given the data in Figure 3, calculate the net gravita-
tional force on the Moon due to the gravitational
forces exerted by Earth and the Sun.
Figure 4
A conical pendulum
to Earth (5.98 ⴛ 1024 kg)
(a) Describe how you would investigate what factors
affect the frequency of revolution of the
pendulum.
(b) Draw an FBD of the bob at the instant shown.
What force causes the centripetal acceleration?
r = 3.84 ⴛ 105 km
r = 1.49 ⴛ 108 km
to Sun
Moon
(1.99 ⴛ 1030 kg)
(7.35 ⴛ 1022 kg)
160 Chapter 3
Figure 3
NEL
Unit 1
(c) Calculate the speed of the bob.
(d) Determine the frequency of the bob.
teacher
rope
23. How would the device in Figure 4 of the Chapter 3
introductory Try This Activity have to be modified to
make it work aboard the ISS? Explain your answer.
24 m
24 m
Making Connections
24. In April 2001, an automobile race on a 2.4-km oval
racetrack in Fort Worth, Texas, was cancelled because
of complaints of danger by the drivers. At the high
speeds involved, the drivers experienced forces nearly
twice as great as the forces on most racetracks. Find
out why the racetrack was so dangerous and why the
race was cancelled. Explain the physics of the situation, referring to the banking angle of the track and
the net forces on the drivers.
GO
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Extension
25. Obtain the value of g at the surface of Earth using the
motion of the Moon. Assume that the Moon’s period
around Earth is 27 d 8 h and that the radius of its
orbit is 60.1 times the radius (6.38 106 m) of Earth.
24 m
Figure 5
28. A baseball player works out by slugging a baseball in
an Olympic stadium. The ball hangs from a long,
light vertical rod that is free to pivot about its upper
end (P), as shown in Figure 6(a). The ball starts off
with a large horizontal velocity, but the rod pulls it up
in a big vertical circle and it coasts slowly over the top
as shown. If we look at the ball on the way down,
after the rod has swung through 270°, which of the
vector arrows shown in Figure 6(b) gives the correct
direction for the acceleration of the ball? Ignore air
resistance and friction at the pivot.
(a)
P
26. Snoopy, in hot pursuit of the Red Baron, is flying his
vintage warplane in a “loop-the-loop” path. His
instruments tell him that the plane is level (at the
bottom of the loop) and travelling at a speed of
180 km/h. He is sitting on a set of bathroom scales.
He notes that the scales show four times his normal
weight. What is the radius of the loop, in metres?
down
Sir Isaac Newton Contest Question
27. Your favourite physics teacher who is late for class
attempts to swing from the roof of a 24-m high
building to the bottom of an identical building using
a 24-m rope as shown in Figure 5. She starts from rest
with the rope horizontal, but the rope will break if the
tension force in it is twice the weight of the teacher.
How high is the swinging physicist above level when
the rope breaks? (Hint: Apply the law of conservation
of energy.)
Sir Isaac Newton Contest Question
(b)
P
270º
ball
E
A
B
C
D
start
Figure 6
Sir Isaac Newton Contest Question
NEL
Circular Motion 161