Unit Nine Circular Motion
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Transcript Unit Nine Circular Motion
Circular Motion and Gravitation
Section 1
Preview
Section 1 Circular Motion
Section 2 Newton’s Law of Universal Gravitation
Section 3 Motion in Space
Section 4 Torque and Simple Machines
Circular Motion and Gravitation
Section 1
What do you think?
• Consider the following objects moving in circles:
•
•
•
•
A car traveling around a circular ramp on the highway
A ball tied to a string being swung in a circle
The moon as it travels around Earth
A child riding rapidly on a playground merry-go-round
• For each example above, answer the following:
• Is the circular motion caused by a force?
• If so, in what direction is that force acting?
• What is the source of the force acting on each object?
Circular Motion and Gravitation
Tangential Speed (vt)
• Speed in a direction tangent to the
circle
• Uniform circular motion: vt has a
constant value
– Only the direction changes
– Example shown to the right
• How would the tangential speed of
a horse near the center of a
carousel compare to one near the
edge? Why?
Section 1
Circular Motion and Gravitation
Centripetal Acceleration (ac)
• Acceleration is a change in
velocity (size or direction).
• Direction of velocity changes
continuously for uniform circular
motion.
• What direction is the acceleration?
– the same direction as v
– toward the center of the circle
• Centripetal means “center
seeking”
Section 1
Circular Motion and Gravitation
Section 1
Centripetal Acceleration (magnitude)
• How do you think the magnitude of the acceleration
depends on the speed?
• How do you think the magnitude of the acceleration
depends on the radius of the circle?
Circular Motion and Gravitation
Section 1
Tangential Acceleration
• Occurs if the speed increases
• Directed tangent to the circle
• Example: a car traveling in a circle
– Centripetal acceleration maintains the circular motion.
• directed toward center of circle
– Tangential acceleration produces an increase or
decrease in the speed of the car.
• directed tangent to the circle
Circular Motion and Gravitation
Centripetal Acceleration
Click below to watch the Visual Concept.
Visual Concept
Section 1
Circular Motion and Gravitation
Centripetal Force (Fc)
Fc mac
vt 2
and ac
r
mvt 2
so Fc
r
Section 1
Circular Motion and Gravitation
Centripetal Force
• Maintains motion in a circle
• Can be produced in different
ways, such as
– Gravity
– A string
– Friction
• Which way will an object
move if the centripetal force
is removed?
– In a straight line, as shown on
the right
Section 1
Circular Motion and Gravitation
Section 1
Describing a Rotating System
• Imagine yourself as a passenger in a car turning quickly
to the left, and assume you are free to move without the
constraint of a seat belt.
– How does it “feel” to you during the turn?
– How would you describe the forces acting on you during this
turn?
• There is not a force “away from the center” or “throwing
you toward the door.”
– Sometimes called “centrifugal force”
• Instead, your inertia causes you to continue in a straight
line until the door, which is turning left, hits you.
Circular Motion and Gravitation
Section 1
Classroom Practice Problems
• A 35.0 kg child travels in a circular path with a
radius of 2.50 m as she spins around on a
playground merry-go-round. She makes one
complete revolution every 2.25 s.
– What is her speed or tangential velocity? (Hint: Find
the circumference to get the distance traveled.)
– What is her centripetal acceleration?
– What centripetal force is required?
• Answers: 6.98 m/s, 19.5 m/s2, 682 N
Circular Motion and Gravitation
Section 1
Now what do you think?
• Consider the following objects moving in circles:
•
•
•
•
A car traveling around a circular ramp on the highway
A ball tied to a string being swung in a circle
The moon as it travels around Earth
A child riding rapidly on a playground merry-go-round
• For each example above, answer the following:
• Is the circular motion caused by a force?
• If so, in what direction is that force acting?
• What is the source of the force acting on each object?
Circular Motion and Gravitation
Section 2
What do you think?
Imagine an object hanging from a spring scale.
The scale measures the force acting on the
object.
• What is the source of this force? What is pulling or
pushing the object downward?
• Could this force be diminished? If so, how?
• Would the force change in any way if the object was
placed in a vacuum?
• Would the force change in any way if Earth stopped
rotating?
Circular Motion and Gravitation
Section 2
Newton’s Thought Experiment
• What happens if you fire a
cannonball horizontally at
greater and greater speeds?
• Conclusion: If the speed is
just right, the cannonball will
go into orbit like the moon,
because it falls at the same
rate as Earth’s surface
curves.
• Therefore, Earth’s
gravitational pull extends to
the moon.
Circular Motion and Gravitation
Section 2
Law of Universal Gravitation
• Fg is proportional to the product of the masses (m1m2).
• Fg is inversely proportional to the distance squared (r2).
– Distance is measured center to center.
• G converts units on the right (kg2/m2) into force units (N).
– G = 6.673 x 10-11 N•m2/kg2
Circular Motion and Gravitation
Law of Universal Gravitation
Section 2
Circular Motion and Gravitation
Section 2
Gravitational Force
• If gravity is universal and exists between all
masses, why isn’t this force easily observed in
everyday life? For example, why don’t we feel a
force pulling us toward large buildings?
– The value for G is so small that, unless at least one of
the masses is very large, the force of gravity is
negligible.
Circular Motion and Gravitation
Ocean Tides
•
•
•
•
What causes the tides?
How often do they occur?
Why do they occur at certain times?
Are they at the same time each day?
Section 2
Circular Motion and Gravitation
Section 2
Ocean Tides
• Newton’s law of universal gravitation is used to explain
the tides.
– Since the water directly below the moon is closer than
Earth as a whole, it accelerates more rapidly toward
the moon than Earth, and the water rises.
– Similarly, Earth accelerates more rapidly toward the
moon than the water on the far side. Earth moves
away from the water, leaving a bulge there as well.
– As Earth rotates, each location on Earth passes
through the two bulges each day.
– Link to web
Circular Motion and Gravitation
Section 2
Gravity is a Field Force
• Earth, or any other mass,
creates a force field.
• Forces are caused by an
interaction between the
field and the mass of the
object in the field.
• The gravitational field (g)
points in the direction of
the force, as shown.
Circular Motion and Gravitation
Calculating the value of g
• Since g is the force acting on a 1 kg object, it
has a value of 9.81 N/m (on Earth).
– The same value as ag (9.81 m/s2)
• The value for g (on Earth) can be calculated
as shown below.
Fg
GmmE GmE
g
2
2
m
mr
r
Section 2
Circular Motion and Gravitation
Section 2
Classroom Practice Problems
• Find the gravitational force that Earth
(mE = 5.97 1024 kg) exerts on the moon
(mm= 7.35 1022 kg) when the distance between
them is 3.84 x 108 m.
– Answer: 1.99 x 1020 N
• Find the strength of the gravitational field at a
point 3.84 x 108 m from the center of Earth.
– Answer: 0.00270 N/m or 0.00270 m/s2
Circular Motion and Gravitation
Section 2
Now what do you think?
Imagine an object hanging from a spring scale.
The scale measures the force acting on the
object.
– What is the source of this force? What is pulling or
pushing the object downward?
– Could this force be diminished? If so, how?
– Would the force change in any way if the object was
placed in a vacuum?
– Would the force change in any way if Earth stopped
rotating?
Circular Motion and Gravitation
Section 3
What do you think?
• Make a sketch showing the path of Earth as it
orbits the sun.
• Describe the motion of Earth as it follows this
path.
• Describe the similarities and differences
between the path and motion of Earth and that
of other planets.
Circular Motion and Gravitation
Section 3
What do you think?
• What does the term weightless mean to you?
• Have you ever observed someone in a
weightless environment? If so, when?
• How did their weightless environment differ from a
normal environment?
Circular Motion and Gravitation
Section 3
Weight and Weightlessness
• Bathroom scale
– A scale measures the downward force exerted on it.
– Readings change if someone pushes down or lifts up
on you.
• Your scale reads the normal force acting on you.
Circular Motion and Gravitation
Section 3
Apparent Weightlessness
• Elevator at rest: the scale reads the weight (600 N).
• Elevator accelerates downward: the scale reads less.
• Elevator in free fall: the scale reads zero because it no
longer needs to support the weight.
Circular Motion and Gravitation
Section 3
Apparent Weightlessness
• You are falling at the same rate as your
surroundings.
– No support force from the floor is needed.
• Astronauts are in orbit, so they fall at the same
rate as their capsule.
• True weightlessness only occurs at great
distances from any masses.
– Even then, there is a weak gravitational force.
Circular Motion and Gravitation
Section 3
Now what do you think?
• Make a sketch showing the path of Earth as it
orbits the sun.
• Describe the motion of Earth as it follows this
path.
• Describe the similarities and differences
between the path and motion of Earth and that
of other planets.
Circular Motion and Gravitation
Section 3
Now what do you think?
• What does the term weightless mean to you?
• Have you ever observed someone in a
weightless environment? If so, when?
• How did their weightless environment differ from a
normal environment?
Circular Motion and Gravitation
Section 4
Simple Machines
• Change the size or direction of the input force
• Mechanical advantage (MA) compares the input
force to the output force.
– When Fout > Fin then MA > 1
• MA can also be determined from the distances
the input and output forces move.
Fout
din
MA
Fin dout
Circular Motion and Gravitation
Overview of Simple Machines
Click below to watch the Visual Concept.
Visual Concept
Section 4
Circular Motion and Gravitation
Section 4
Simple Machines
• Simple machines alter the force
and the distance moved.
• For the inclined plane shown:
– F2 < F1 so MA >1 and d2 > d1
• If the ramp is frictionless, the
work is the same in both cases.
– F1d1 = F2d2
• With friction, F2d2 > F1d1.
– The force is reduced but the work
done is greater.
Circular Motion and Gravitation
Section 4
Efficiency of Simple Machines
• Efficiency measures work output compared to
work input.
– In the absence of friction, they are equal.
• Real machines always have efficiencies less
than 1, but they make work easier by changing
the force required to do the work.
Wout
eff
Win
Circular Motion and Gravitation
Preview
• Multiple Choice
• Short Response
• Extended Response
Section 4
Circular Motion and Gravitation
Section 4
Multiple Choice
1. An object moves in a circle at a constant speed.
Which of the following is not true of the object?
A. Its acceleration is constant.
B. Its tangential speed is constant.
C. Its velocity is constant.
D. A centripetal force acts on the object.
Circular Motion and Gravitation
Section 4
Multiple Choice
1. An object moves in a circle at a constant speed.
Which of the following is not true of the object?
A. Its acceleration is constant.
B. Its tangential speed is constant.
C. Its velocity is constant.
D. A centripetal force acts on the object.
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
Use the passage below to answer questions 2–
3.
A car traveling at 15 m/s on a flat surface turns in a
circle with a radius of 25 m.
2. What is the centripetal acceleration of the car?
F. 2.4 10-2 m/s2
G. 0.60 m/s2
H. 9.0 m/s2
J. zero
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
Use the passage below to answer questions 2–
3.
A car traveling at 15 m/s on a flat surface turns in a
circle with a radius of 25 m.
2. What is the centripetal acceleration of the car?
F. 2.4 10-2 m/s2
G. 0.60 m/s2
H. 9.0 m/s2
J. zero
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
Use the passage below to answer questions 2–
3.
A car traveling at 15 m/s on a flat surface turns in a
circle with a radius of 25 m.
3. What is the most direct cause of the car’s
centripetal acceleration?
A. the torque on the steering wheel
B. the torque on the tires of the car
C. the force of friction between the tires and the
road
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
Use the passage below to answer questions 2–
3.
A car traveling at 15 m/s on a flat surface turns in a
circle with a radius of 25 m.
3. What is the most direct cause of the car’s
centripetal acceleration?
A. the torque on the steering wheel
B. the torque on the tires of the car
C. the force of friction between the tires and the
road
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
4. Earth (m = 5.97 1024 kg) orbits the sun (m =
1.99 1030 kg) at a mean distance of 1.50
1011 m. What is the gravitational force of the sun
on Earth? (G = 6.673 10-11 N•m2/kg2)
F. 5.29 1032 N
G. 3.52 1022 N
H. 5.90 10–2 N
J. 1.77 10–8 N
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
4. Earth (m = 5.97 1024 kg) orbits the sun (m =
1.99 1030 kg) at a mean distance of 1.50
1011 m. What is the gravitational force of the sun
on Earth? (G = 6.673 10-11 N•m2/kg2)
F. 5.29 1032 N
G. 3.52 1022 N
H. 5.90 10–2 N
J. 1.77 10–8 N
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
5. Which of the following is a correct
mE
gG
g the
interpretationaof
expression
r2
?
A. Gravitational field strength changes with
an
object’s distance from Earth.
B. Free-fall acceleration changes with an
object’s
distance from Earth.
C. Free-fall acceleration is independent of the
falling
object’s mass.
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
5. Which of the following is a correct
mE
gG
g the
interpretationaof
expression
r2
?
A. Gravitational field strength changes with
an
object’s distance from Earth.
B. Free-fall acceleration changes with an
object’s
distance from Earth.
C. Free-fall acceleration is independent of the
falling
object’s mass.
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
6. What data do you need to calculate the orbital
speed of a satellite?
F. mass of satellite, mass of planet, radius of
orbit
G. mass of satellite, radius of planet, area of
orbit
H. mass of satellite and radius of orbit only
J. mass of planet and radius of orbit only
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
6. What data do you need to calculate the orbital
speed of a satellite?
F. mass of satellite, mass of planet, radius of
orbit
G. mass of satellite, radius of planet, area of
orbit
H. mass of satellite and radius of orbit only
J. mass of planet and radius of orbit only
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
7. Which of the following choices correctly
describes the orbital relationship between Earth
and the sun?
A. The sun orbits Earth in a perfect circle.
B. Earth orbits the sun in a perfect circle.
C. The sun orbits Earth in an ellipse, with Earth
at one focus.
D. Earth orbits the sun in an ellipse, with the sun
at one focus.
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
7. Which of the following choices correctly
describes the orbital relationship between Earth
and the sun?
A. The sun orbits Earth in a perfect circle.
B. Earth orbits the sun in a perfect circle.
C. The sun orbits Earth in an ellipse, with Earth
at one focus.
D. Earth orbits the sun in an ellipse, with the sun
at one focus.
Circular Motion and Gravitation
Multiple Choice, continued
Use the diagram to
answer
questions
8–9. acting on
8.
The three forces
the wheel have equal
magnitudes. Which force will
produce the greatest torque on the wheel?
F. F1
G. F2
H. F3
J. Each force will produce the same torque.
Section 4
Circular Motion and Gravitation
Multiple Choice, continued
Use the diagram to
answer
questions
8–9. acting on
8.
The three forces
the wheel have equal
magnitudes. Which force will
produce the greatest torque on the wheel?
F. F1
G. F2
H. F3
J. Each force will produce the same torque.
Section 4
Circular Motion and Gravitation
Multiple Choice, continued
Use the diagram to
answer
questions
8–9.
9.
If each force
is 6.0 N, the
angle between F1 and F2 is
60.0°, and the radius of the
wheel is 1.0 m, what is the
resultant torque on the wheel?
A. –18 N•m
B. –9.0 N•m
C. 9.0 N•m
D. 18 N•m
Section 4
Circular Motion and Gravitation
Multiple Choice, continued
Use the diagram to
answer
questions
8–9.
9.
If each force
is 6.0 N, the
angle between F1 and F2 is
60.0°, and the radius of the
wheel is 1.0 m, what is the
resultant torque on the wheel?
A. –18 N•m
B. –9.0 N•m
C. 9.0 N•m
D. 18 N•m
Section 4
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
10. A force of 75 N is applied to a lever. This force
lifts a
load weighing 225 N. What is the
mechanical
advantage of the lever?
F. 1/3
G. 3
H. 150
J. 300
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
10. A force of 75 N is applied to a lever. This force
lifts a
load weighing 225 N. What is the
mechanical
advantage of the lever?
F. 1/3
G. 3
H. 150
J. 300
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
11. A pulley system has an efficiency of 87.5
percent.
How much work must you do to
lift a desk weighing 1320 N to a height of 1.50
m?
A. 1510 J
B. 1730 J
C. 1980 J
D. 2260 J
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
11. A pulley system has an efficiency of 87.5
percent.
How much work must you do to
lift a desk weighing 1320 N to a height of 1.50
m?
A. 1510 J
B. 1730 J
C. 1980 J
D. 2260 J
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
12. Which of the following statements is correct?
F. Mass and weight both vary with location.
G. Mass varies with location, but weight does
not.
H. Weight varies with location, but mass does
not.
J. Neither mass nor weight varies with location.
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
12. Which of the following statements is correct?
F. Mass and weight both vary with location.
G. Mass varies with location, but weight does
not.
H. Weight varies with location, but mass does
not.
J. Neither mass nor weight varies with location.
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
13. Which astronomer discovered that planets
travel in elliptical rather than circular orbits?
A. Johannes Kepler
B. Nicolaus Copernicus
C. Tycho Brahe
D. Claudius Ptolemy
Circular Motion and Gravitation
Section 4
Multiple Choice, continued
13. Which astronomer discovered that planets
travel in elliptical rather than circular orbits?
A. Johannes Kepler
B. Nicolaus Copernicus
C. Tycho Brahe
D. Claudius Ptolemy
Circular Motion and Gravitation
Section 4
Short Response
14. Explain how it is possible for all the water to
remain in a pail that is whirled in a vertical
path, as shown
below.
Circular Motion and Gravitation
Section 4
Short Response
14. Explain how it is possible for all the water to
remain in a pail that is whirled in a vertical
path, as shown
below.
Answer: The water
remains in the pail even
when the pail is upside
down because the water
tends to move in a
straight path due to
inertia.
Circular Motion and Gravitation
Section 4
Short Response, continued
15. Explain why approximately two high tides take
place
every day at a given location on Earth.
Circular Motion and Gravitation
Section 4
Short Response, continued
15. Explain why approximately two high tides take
place
every day at a given location on Earth.
Answer: The moon’s tidal forces create two
bulges on Earth. As Earth rotates on its axis
once per day, any given point on Earth passes
through both bulges.
Circular Motion and Gravitation
Section 4
Short Response, continued
16. If you used a machine to increase the output
force,
what factor would have to be
sacrificed? Give an
example.
Circular Motion and Gravitation
Section 4
Short Response, continued
16. If you used a machine to increase the output
force,
what factor would have to be
sacrificed? Give an
example.
Answer: You would have to apply the input force
over a greater distance. Examples may include
any machines that increase output force at the
expense of input distance.
Circular Motion and Gravitation
Section 4
Extended Response
17. Mars orbits the sun (m = 1.99 1030 kg) at a
mean
distance of 2.28 1011 m. Calculate
the length of
the Martian year in Earth
days. Show all of your
work. (G =
6.673 10–11 N•m2/kg2)
Circular Motion and Gravitation
Section 4
Extended Response
17. Mars orbits the sun (m = 1.99 1030 kg) at a
mean
distance of 2.28 1011 m. Calculate
the length of
the Martian year in Earth
days. Show all of your
work. (G =
6.673 10–11 N•m2/kg2)
Answer: 687 days