Download Modified True/False Indicate whether the sentence or statement is

Modified True/False
Indicate whether the sentence or statement is true or false. If false, change the identified word or phrase to make the
sentence or statement true.
____
1. For an object travelling with uniform circular motion, its acceleration is always directed tangent to the circle.
___________________________________
____
2. When a curve in the road is banked, a component of the normal force contributes to the centripetal force.
______________________________
____
3. For an object travelling with uniform circular motion, its instantaneous velocity is always directed toward the
centre of the circle. ______________________________
____
4. For an object travelling with uniform circular motion, the centripetal force is always directed radially outward
from the centre of the circle. ___________________________________
____
5. A rock is tied to a string and whirled around in a circle that describes a vertical plane. The tension in the
string is greatest at the bottom of the circle and least at the top. ______________________________
____
6. When riding a merry-go-round, the centripetal force that acts inward is the only real force acting on a person.
The centrifugal force does not really exist, but is used to account for the “apparent” force that acts on the
person as viewed from the noninertial frame of reference. _________________________
____
7. Centrifugal forces and Coriolis forces are both fictitious forces used to explain motion in rotating frames of
reference. _________________________
____
8. Of Newton’s three laws of motion, Newton’s law of universal gravitation is most closely associated with his
second law of motion. ______________________________
____
9. If Earth’s mass and radius were both one half their present values, the acceleration due to gravity on the
surface of the Earth would be four times its present value. __________________________________
____ 10. All geosynchronous satellites must have the same orbital radius. _________________________
Multiple Choice
Identify the letter of the choice that best completes the statement or answers the question.
____ 11. For an object travelling with “uniform circular motion,”
a. its velocity is constant
b. its acceleration is always directed tangent to the circle
c. its velocity is always directed toward the centre of the circle
d. its speed and distance from the centre of the circle are constant
e. its speed may change provided the radius of the circle is constant
____ 12. For an object travelling with “uniform circular motion,” its acceleration is
a. zero because the speed is constant
b. directed tangent to the circle
c. directed toward the centre of the circle
d. changing in magnitude depending on its position in the circle
e. directed outward from the centre of the circle
____ 13. An object moves with a speed of 2.4 m/s in a circle of radius 1.6 m. Its centripetal acceleration is
a. 9.2 m/s2
d. 1.5 m/s2
2
b. 7.4 m/s
e. 0.94 m/s2
c. 3.6 m/s2
____ 14. A child whirls a ball around in circles on the end of a 48 cm long string at a frequency of 2.5 Hz. What is the
ball’s centripetal acceleration?
d. 38 m/s2
a. 1.2 × 104 m/s2
b. 1.2 × 102 m/s2
e. 3.0 m/s2
c. 47 m/s2
____ 15. A passenger on a Ferris wheel of diameter 22 m makes one complete revolution every 45 s. What is the
passenger’s centripetal acceleration?
d. 0.21 m/s2
a. 19 m/s2
2
b. 13 m/s
e. 0.068 m/s2
2
c. 0.43 m/s
____ 16. A fighter jet flies at 328 km/h in an arc of radius 235 m. How many “g’s” of centripetal acceleration does the
pilot experience? (1g = 9.8 m/s2)
a. 47
d. 3.6
b. 35
e. 1.5
c. 3.8
____ 17. A car drives around a corner in a radius of 16 m. Passengers experience a centripetal acceleration of 4.9 m/s2.
What is the car’s speed expressed in kilometres per hour?
a. 42
d. 12
b. 32
e. 2
c. 22
____ 18. A wheel of diameter 85 cm spins at a rate such that a point on the rim of the wheel has an acceleration of 45
m/s2. How many rotations does the wheel make in 1.0 min?
a. 1.7 × 102
d. 6.9
b. 98
e. 0.93
c. 69
____ 19. The diagram below shows a rock on the end of a string being whirled around in a circle in the horizontal
plane. The motion is viewed from above and the stone is rotating clockwise. What are the directions
associated with its instantaneous velocity and instantaneous acceleration, respectively?
a. south, east
d. west, west
b. east, west
e. south, west
c. south, south
____ 20. The diagram below illustrates a car travelling with a constant speed around a curve in the road. The curve is
banked. Which force represents the direction of the net force acting on the car?
____ 21.
____ 22.
____ 23.
____ 24.
____ 25.
____ 26.
d. F4
a. F1
b. F2
e. F5
c. F3
The reason that curves on roads are often banked is because
a. the coefficient of static friction is increased
b. the coefficient of kinetic friction is increased
c. a component of the normal force can contribute to the centripetal force
d. the gravitational force acting on the car is reduced
e. the normal force acting on the car is reduced
A rock is tied to the end of a string and whirled around in a circle that describes a vertical plane. At which
position is the tension in the string the least?
a. at the bottom of the circle
b. at the top of the circle
c. on the ascending side of the circle
d. on the descending side of the circle
e. The tension in the string is the same at all places on the circle.
A rock is tied to the end of a 35 cm long string and whirled around in a circle that describes a vertical plane.
The tension in the string becomes zero when the speed of the rock is
a. 9.8 × 102 cm/s
d. 9.8 cm/s
b. 1.9 × 102 cm/s
e. 1.9 cm/s
c. 19 cm/s
Imagine you are a passenger upside-down at the top of a vertical looping roller coaster. The centripetal force
acting on you at this position
a. is perhaps the least of anywhere in the loop
b. is supplied at least partly by gravity
c. is supplied partly by the seat of the roller coaster
d. is directed vertically downward
e. all of the above
Two identical masses, A and B, are each tied to the ends of strings A and B, respectively, and whirled around
in circles that are both oriented horizontally. The length of string A is twice that of string B. For the tension in
the strings to be the same, the ratio of the speeds of the masses (vA:vB) must be
a. 1.4:1
d. 1:2
b. 1:1.4
e. 1:1
c. 2:1
A 1.0-kg and a 2.0-kg mass are each tied to the ends of identical strings and whirled around in circles that
describe a horizontal plane. The larger mass moves with a speed of 3.2 m/s. For the tension in the two strings
to be the same, the smaller mass must be moving with a speed of
a. 6.4 m/s
d. 2.3 m/s
b. 4.5 m/s
e. 1.6 m/s
c. 3.2 m/s
____ 27. For objects travelling with uniform circular motion, the centrifugal force they experience
a. is radially outward
b. is apparently present in the noninertial frame of reference
c. increases with speed
d. doesn’t actually exist
e. all of the above
____ 28. The acceleration due to gravity on the surface of a planet having twice the Earth’s mass and twice its radius
would be
d. 4.9 m/s2
a. 39.2 m/s2
2
b. 19.6 m/s
e. 2.45 m/s2
2
c. 9.8 m/s
____ 29. Planet X has a radius 4 times that of Earth and the acceleration due to gravity at the surface of planet X is 4.9
m/s2. The mass of Planet X compared to Earth’s mass is
a. 16 times
d. 2 times
b. 8 times
e. the same
c. 4 times
____ 30. Planet X has a mass 8 times that of Earth and the acceleration due to gravity at its surface is 19.6 m/s2. The
radius of planet X compared to Earth is
a. 16 times
d. 2 times
b. 8 times
e. the same
c. 4 times
____ 31. With all other things being equal, had the value of the universal gravitational constant been twice its present
value, your weight would be
a. four times as great
d. one-half as great
b. two times as great
e. one-quarter as great
c. the same
____ 32. The person responsible for determining the value of the universal gravitational constant is
a. Galileo Galilei
d. Henry Cavendish
b. Isaac Newton
e. Albert Einstein
c. William Herschel
____ 33. The force of gravity acting on a 10-kg object at an altitude equivalent to the Earth’s radius is
a. 49 N
d. 5.0 N
b. 24 N
e. 2.4 N
c. 9.8 N
____ 34. The orbital radius of a satellite circling the Earth
a. depends only on its orbital speed
b. depends only on its mass
c. depends on both its orbital speed and its mass
d. is the same for all satellites
e. is directly proportional to its orbital speed
____ 35. The orbital speed of a satellite at an altitude equivalent to Earth’s radius (rE = 6.38 × 106 m) is (mE = 5.98 ×
1024 kg, G = 6.67 × 10–11 N·m2/kg2)
a. 9.8 × 103 m/s
d. 4.9 × 103 m/s
3
b. 7.9 × 10 m/s
e. 2.5 × 103 m/s
c. 5.6 × 103 m/s
____ 36. Astronauts on board an orbiting space station appear to be “floating” because
a. they are in the vacuum of space
b. they are outside Earth’s gravitational influence
c. the force of gravity acting on them has been reduced to an insignificant level
d. they have become truly “weightless”
e. they are in free fall along with the space station itself
____ 37. To produce an artificial gravity on board a space station
a. would be impossible
b. would require an enormous quantity of matter
c. is easily achieved by rotating the space station
d. would be possible by maintaining an inertial frame of reference
e. is purely science fiction
____ 38. The relationship between the gravitational force of attraction, FG, of two objects on one another and their
separation distance r is best illustrated by which of the following graphs?
a. A
d. D
b. B
e. E
c. C
____ 39. Which of the following graphs best illustrates the relationship between a satellite’s orbital radius ro and its
orbital speed vo?
a. A
d. D
b. B
e. E
c. C
____ 40. Which of the following graphs best illustrates the relationship between a satellite’s orbital speed vo and its
orbital radius ro?
a. A
b. B
c. C
Completion
Complete each sentence or statement.
d. D
e. E
41. For an object travelling with uniform circular motion, the direction of its acceleration at any point is
______________________________.
42. For an object travelling with uniform circular motion, the direction of its instantaneous velocity at any point is
_________________________.
43. Besides static friction, the force responsible for directing a car around a banked curve is the
____________________ force.
44. A rock is tied to the end of a string and whirled around in a circle that describes a vertical plane. The position
of the stone when the tension in the string is the greatest is ______________________________. The position
of the stone when the tension in the string is the least is ______________________________.
45. For a planet orbiting the Sun, the centripetal force is supplied by ____________________.
46. The law of universal gravitation states that the force of gravitational attraction between two objects is directly
proportional to ___________________________________ and inversely proportional to
________________________________________.
47. The physicist ____________________ was responsible for the theory of universal gravitation, but it was
____________________ who determined the value of the universal gravitational constant.
48. If Earth’s radius and mass were both twice their present values, the acceleration due to gravity at Earth’s
surface would have a value of ____________________.
49. A satellite in geosynchronous orbit has an orbital period of ____________________.
50. The speed of a satellite in orbit is determined by the ____________________ of the orbit.
Matching
The diagram below shows an object travelling with uniform circular motion as viewed from above. Match
each of the following quantities with its proper direction.
a. north
b. south
____
____
____
____
51.
52.
53.
54.
c. east
d. west
instantaneous velocity
centripetal acceleration
centripetal force
centrifugal force
Match each of the following situations with the type of force responsible for the object’s centripetal
acceleration.
a. gravity
d. static friction
b. tension
e. kinetic friction
c. normal
____ 55. a planet orbiting the Sun
____ 56. a car driving around an icy banked curve
____ 57. a car driving around a flat curve
Short Answer
58. For objects travelling with uniform circular motion, what are the directions of the instantaneous velocity and
acceleration at any point on the circle?
59. Why does the centripetal force not appear on any free-body diagrams.
60. Starting with the expression
, derive the other two expressions for centripetal acceleration.
61. Why are curves in roadways often banked?
62. The free-body diagrams of an object sliding down a ramp and a car travelling on a banked curve are very
similar. Why are the analyses of the forces in these two situations very different?
63. Why do roller coasters use clothoid loops instead of circular loops?
64. Explain why “centrifugal” forces are fictitious forces.
65. Using three different examples, show that the centripetal force is always supplied by one or a combination of
forces.
66. Sketch two graphs. The first graph should illustrate the relationship between the force of gravity Earth exerts
on objects and the mass of the objects. The second graph should show the variation of the force of gravity a
pair of objects exert on one another and the separation distance between the objects’ centres of mass.
67. Describe how the acceleration of gravity on Earth’s surface would be different if both its mass and radius
were twice their present values. Provide reasons for your answer.
68. Describe the limitations of the law of universal gravitation.
69. Describe how the law of universal gravitation is closely associated with Newton’s third law of motion.
70. Given the following quantities, show that the gravitational field strength of Earth at its surface is 9.8 N/kg.
mE = 5.98 × 1024 kg, rE = 6.38 × 106 m, G = 6.67 × 10–11 N·m2/kg2
71. Describe what is meant by the term “apparent weightlessness” as it applies to astronauts on board an orbiting
space station. As well, provide the proper explanation of what is really happening.
72. Describe how “artificial” gravity can be produced on board an orbiting space station. Discuss the physics of
the situation.
Problem
73. On a midway ride called The Round Up, participants climb into a cylindrical apparatus and stand upright with
their backs against the wall of the cylinder. The apparatus begins to spin, first in the horizontal plane and then
tipping into the vertical plane so that the riders’ bodies are parallel to the ground below. Consider a rider of
mass 60.0 kg. The radius of the apparatus is 8.0 m and it spins with a period of 4.0 s.
(a) Determine the centripetal force acting on the person when the apparatus is still in the horizontal plane.
Provide the appropriate free-body diagram and state the force that supplies the centripetal force in this
situation.
(b) Draw the free-body diagram of the person at the bottom of the apparatus when spinning in the vertical
plane and determine how heavy the person feels at that position (i.e., what is the value of the normal force at
that position?).
(c) What minimum speed must the person be moving with in order to keep from falling away from the side of
the cylinder when rotating in the vertical plane? State where in the circle this would occur and draw the
appropriate free-body diagram.
74. A 2.0 × 102-g mass is tied on the end of a 1.6 m long string and whirled around in a circle that describes a
vertical plane.
(a) What is the minimum frequency of rotation required to keep the mass moving in a circle?
(b) Calculate the maximum tension in the string at this frequency.
75. A 0.50-g insect rests on a compact disc at a distance of 4.0 cm from the centre. The disc’s rate of rotation
varies from 3.5 Hz to 8.0 Hz in order to maintain a constant data sampling rate.
(a) What are the insect’s minimum and maximum centripetal accelerations during its rotation around the disc?
(b) What is the minimum value of the coefficient of static friction that would prevent the insect from slipping
off the disc at the slowest rotation rate?
76. A pilot of mass 75 kg takes her plane into a dive, pulling out of it along a circular arc as she nears the ground.
If the plane is flying at 1.5 × 102 km/h along the arc, what is its radius such that the pilot feels four times
heavier than normal? Provide an appropriate free-body diagram.
77. A rock of mass 4.0 × 102 g is tied to one end of a string that is 2.0 m in length. Holding the other end above
his head, a boy swings the rock around in a circle whose plane is parallel to the ground.
(a) If the string can withstand a maximum tension of 4.5 N before breaking, what angle to the vertical does
the string reach just before breaking?
(b) At what speed is the rock travelling just as the string breaks?
78. A pilot of mass 60.0 kg is flying her plane in a vertically oriented circular loop. Just at the bottom of the loop,
the plane’s speed is 1.8 × 102 km/h and the pilot feels exactly four times as heavy as she normally does.
(a) What is the radius of the loop?
(b) At what speed must she be flying at the top of the loop in order to feel weightless?
79. An object of mass 6.0 kg is whirled around in a vertical circle on the end of a 1.0 m long string with a
constant speed of 8.0 m/s. Include a free-body diagram for each of the following questions:
(a) Determine the maximum tension in the string, indicating the position of the object at the time the
maximum tension is achieved.
(b) What is the minimum speed the object could be rotated with and maintain a circular path?
(c) If the object is rotated with the same speed (8.0 m/s) on a horizontal surface, what is the tension in the
string if the string is parallel to the surface?
80. A 1.2-kg mass is tied on the end of a 80.0 cm long rope and whirled around in a vertically-oriented circle at
an ever-increasing rate. The rope breaks just as the frequency of rotation reaches 8.4 Hz. What maximum
mass could this rope support if it is simply held vertically? Provide an appropriate free-body diagram.
81. A flea stands on the end of a 1.0 cm long sweep second hand of a clock that rests horizontally on a table.
What is the minimum coefficient of static friction which would allow the flea to stay there without slipping?
Include an appropriate free-body diagram.
82. A ball of mass 4.0 kg is attached to the end of a 1.2 m long string and whirled around in a circle that describes
a vertical plane.
(a) What is the minimum speed that the ball can be moving at and still maintain a circular path? Provide a
free-body diagram.
(b) At this speed, what is the maximum tension in the string? Provide another free-body diagram.
(c) If the ball is rotated in a horizontal circle at the same speed with the end of the string held above the head,
what angle does the string make with the horizontal?
83. What force does Earth exert on a 80.0-kg astronaut at an altitude equivalent to 2.5 times Earth’s radius?
84. A planet has a mass of 2.5 times that of Earth and a radius 1.2 times Earth’s radius. How much would a 60.0kg person weigh at the planet’s surface?
85. The gravitational field strength at the surface of a planet is 3.4 N/kg. If the planet’s mass is 7.2 × 1022 kg,
what is its radius?
86. A satellite orbits Earth at an altitude of 325 km above the planet’s surface. What is its orbital period? Express
your answer in minutes. (rE = 6.38 × 106 m, ME = 5.98 × 1024 kg)
87. An Earth satellite has an orbital period of 3.2 h. What is its orbital radius?
(ME = 5.98 × 1024 kg)
88. A satellite has an orbital speed of 4.2 × 103 m/s. What is its altitude above Earth’s surface?
(ME = 5.98 × 1024 kg, rE = 6.38 × 106 m)
Essay
89. Using the principles of uniform circular motion, explain how centrifuges are able to separate materials that
have a variety of densities.
90. Amusement parks often make use of the centrifugal forces that people apparently feel when travelling in
uniform (and nonuniform) circular motion. Provide three distinct examples and examine the physics
associated with each.
91. Discuss the physics associated with geosynchronous satellites from the following perspectives:
- orbital period
- orbital altitude and latitude
- orbital speed
Answer Section
MODIFIED TRUE/FALSE
1. F, toward the centre of the circle
2. T
4. F, inward toward the centre of the circle
5.T
8. F, third law of motion
9. F, two times its present value
3. F, tangent the circle
6. T
7.T
10.
T
MULTIPLE CHOICE
11.
20.
29.
38.
D
B
B
D
```
12. C
21. C
30. D
39. A
``
13. C
22. B
31. B
40. A
14. B
23. B
32. D
15. D
24. E
33. B
16. D
25. A
34. A
17. B
26. B
35. C
18. B
27. E
36. E
19. E
28. D
37. C
COMPLETION
41.
44.
46.
48.
toward the centre of the circle
42. tangent to the circle
43. normal
at the bottom of the circle, at the top of the circle
45. gravity
the product of their masses, the square of their separation distance
47. Isaac Newton, Henry Cavendish
4.9 m/s2
49. 24 h
50. radius
MATCHING
51. B
52. D
53. D
54. C
55. A
56. C
57. D
SHORT ANSWER
58. The instantaneous velocity is always tangent to the circle and the acceleration (centripetal) is always directed
toward the centre of the circle.
59. The centripetal force is the net force in every case. As such, it is comprised of the sum of all the forces that act
on the object. This being the case, as in any free-body diagram, it does not appear on the diagram itself.
61. When the curve is banked, the centripetal force can be supplied not only by static friction as in the case of
unbanked curves, but also by the component of the normal force directed toward the centre of the circle. This
reduces the amount of static friction required to hold the car in the curve.
62. For a ramp question, the forces must be resolved parallel (and perpendicular) to the plane of the ramp since
this will be the direction of the net force and acceleration if there is any. For a banked curve, the centripetal
acceleration (and net force) are directed toward the centre of the circle which the object is moving in. This
direction will typically be horizontal. Forces must be resolved both horizontally (and vertically) in this
situation.
63. The speed of a roller coaster will decrease as it moves up a loop and increase on the way down. For a circular
loop (of constant radius), the centripetal acceleration will decrease steadily as the roller coaster climbs the
loop and increase steadily as it descends. The resulting centripetal force acting on passengers changes
accordingly. The apparent centrifugal force that passengers feel changes as well. Designers of roller coasters
have found that this results in an unpleasant ride for many passengers and may even lead to injury. When the
clothoid loop is used (changing radius), the radius of the loop steadily decreases as the roller coaster ascends
and increases as it descends. When this is combined with the changing speed, the result is a relatively constant
centripetal acceleration (and force), making for a better ride for the passengers.
64. The centrifugal force is used to explain the apparent force that acts on a object travelling with uniform
circular motion. Consider turning a corner in a car. Your inertia would have you continue to move in a
straight line at a constant speed. Instead, the force of static friction of the road on the car’s tires makes the car
move through the curve taking you with it. You feel as though you want to move outward from the centre of
the curve when, in reality, this feeling stems from your inertia to move straight forward although this
direction is constantly changing. The centrifugal force does not really exist. Instead, it is used to account for
the apparent force acting on an object within a noninertial frame of reference.
65. Example 1: A rock is tied on the end of a string and whirled around in a circle in the horizontal plane. The
centripetal force in this case is supplied by the tension in the string.
Example 2: The centripetal force acting on a planet, keeping it in orbit around the Sun, is supplied by gravity.
Example 3: A car rounding a banked curve has two forces that supply the centripetal force: the force of static
friction and the component of the normal force acting toward the centre of the circle.
66. The force of gravity is directly proportional to the mass of the object it is being exerted upon.
The force of gravity varies inversely as the square of the distance between the centres of mass of the two
objects.
67. Since the acceleration due to gravity varies in the same ways as the force of gravity, its value on the surface of
Earth is directly proportional to the mass of Earth and inversely proportional the square of the planet’s radius.
If both these quantities were twice their present values, the acceleration due to gravity would be one-half its
present value. The effect of doubling the mass would double the acceleration, and the effect of doubling the
radius would reduce the acceleration by a factor of four. Combining these effects results in the acceleration
due to gravity being reduced by a factor of two.
68. The law of universal gravitation applies only to two spherical objects (e.g., Earth and the Moon), to two
objects whose centres are separated by a distance much greater than their size, or to a small object and a very
large object (e.g., a person and Earth).
69. The law of universal gravitation discusses the gravitational force between two objects. For example, the force
of gravity Earth exerts on a person is equal in strength and opposite in direction to the force of gravity the
person exerts on Earth. This is also how Newton’s third law addresses the same situation.
70. The gravitational field strength of Earth at its surface is then 9.8 N/kg.
71. The “weightlessness” experienced by orbiting astronauts is only apparent. A person’s weight is defined as the
force of gravity acting on the person. At the altitude of an orbiting space station, an astronaut is still well
within the gravitational field of Earth and is, therefore, far from being truly weightless. Being in orbit is
identical to being in a state of “free fall.” An orbiting space station can be thought of as having been thrown
horizontally initially. As it falls to the ground, the ground also falls away from beneath it (due to the curvature
of Earth). In orbit, the space station and everything on board is continuously falling to Earth, but never
reaching its surface because the surface falls away from beneath it. It would appear that things are weightless
(i.e., “floating” around inside the space station) when they are just all “falling” together.
72. Artificial gravity can be produced by rotating the space station. Imagine the space station designed as a giant
wheel that spins. For people on board, the fictitious centrifugal force would make them feel as though they
were being pulled toward the outside of the wheel. “Up” would be toward the wheel’s hub. What is actually
happening is that the rim of the wheel is pushing the people toward the hub as it overcomes their inertia to
always move tangent to the wheel. The force upward from the rim of the wheel on people’s feet would feel
exactly the same as a downward pull by some gravitational attractor. The strength of the force is dictated by
the wheel’s radius and spin rate.
PROBLEM
73. (a) The centripetal force acting is 1.2 × 103 N
(b)The normal force acting on the rider is 1.8 × 103 N (upward).
(c) The minimum speed is 8.9 m/s.
74. (a) The minimum frequency is 0.39 Hz.
(b) The maximum tension is 3.9 N [up].
75. (a)The insect’s minimum centripetal acceleration is 19 m/s2 and its maximum centripetal accelerations
is 1.0 × 102 m/s2.
(b) The minimum value of the coefficient of static friction is 2.0.
76. The radius of the arc is 59 m.
77. (a) The angle to the vertical is 29°°.
(b)The speed of the rock is 2.3 m/s.
78. (a)The radius of the arc is 85 m.
(b)The pilot’s speed must be 29 m/s.
79. (a)The maximum tension is 4.4 × 102 N [upward].
(b)The minimum speed of rotation is 3.1 m/s.
(c) The tension in the string would be 3.8 × 102 N.
80. The rope could support a mass of 2.7 × 102 kg.
81. The minimum coefficient of static friction is 1.1 × 10–5.
82. (a)The ball must have a minimum speed of 3.4 m/s to stay in a circular path.
(b)The maximum tension in the rope is 78 N.
(c)The string makes an angle of 45°° to the horizontal.
83. Earth exerts a force of 2.6 × 101 N on the astronaut.
84. The person would weigh 1.0 × 103 N at the planet’s surface.
85. The planet’s radius is 1.2 × 106 m.
86. The orbital period is 91.0 min.
87. The orbital radius is 1.1 × 107 m.
88. The altitude above the Earth’s surface is 1.6 × 107 m.
ESSAY
89. Particles of greater density have more mass per unit volume. Greater mass provides more inertia. As the
centrifuge spins, objects tend to always move tangent to the arc of rotation in accordance with Newton’s first
law. Materials with greater inertia will be more successful in moving outward along the radius of the circle.
More dense materials will migrate toward the outside of the centrifuge, whereas less dense materials will tend
to remain closer to the hub.
90. (a) Clothoid loop roller coaster:
Vertical (clothoid) loops on roller coasters have a radius of curvature that decreases as the coaster ascends
into the loop and increases as it descends. The change in radius compensates for the coaster’s change in speed
through the loop, keeping the centripetal acceleration and the force
acting on the patrons
relatively constant. The centrifugal force experienced by the patrons is relatively constant.
(b) Rotating swings:
Swings are suspended and then rotated around a central point. The angle of the suspended swing becomes
more vertical with the speed of rotation. The centripetal force acting on the swing is supplied by the
horizontal component of the tension. As the speed (and centripetal acceleration) increases so must the angle to
the vertical. At a given speed, a rider feels a constant centrifugal force.
(c) Pendulum ride:
Passengers sit in a multi-seat vehicle that is suspended and swung back and forth repeatedly like a giant
pendulum, eventually “going over the top.” Since their speed increases as the passengers swing downward, an
ever-increasing centrifugal force is experienced. As they swing upward, the sensation of being pushed
outward lessens as the vehicle climbs higher and slows down.
91. For a satellite to be geosynchronous, it must have an orbital period of 24 h (to match the rotational period of
Earth). Such satellites must be placed in orbits above the equator—any other location would not satisfy both
"”orbiting” and “geosynchronicity.” A satellite’s centripetal acceleration is supplied by gravity:
Since T is fixed (24 h), then R, the orbital radius (and altitude) are also fixed.
The centripetal force can also be expressed as:
with T and R both fixed, the orbital speed v is also fixed.