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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
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Circular Motion and Gravitation
TEKS
Section 1
The student is expected to:
4C analyze and describe accelerated motion
in two dimensions using equations, including
projectile and circular examples
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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?
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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?
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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”
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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?
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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
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Circular Motion and Gravitation
Centripetal Acceleration
Click below to watch the Visual Concept.
Visual Concept
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Section 1
Circular Motion and Gravitation
Centripetal Force (Fc)
Fc  mac
vt 2
and ac 
r
mvt 2
so Fc 
r
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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
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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.
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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
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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?
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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?
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Circular Motion and Gravitation
TEKS
Section 2
The student is expected to:
3D explain the impacts of the scientific
contributions of a variety of historical and
contemporary scientists on scientific thought
and society
3F express and interpret relationships
symbolically in accordance with accepted
theories to make predictions and solve
problems mathematically, including problems
requiring proportional reasoning and graphical
vector addition
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Circular Motion and Gravitation
TEKS
Section 2
(Continued):
5A research and describe the historical
development of the concepts of gravitational,
electromagnetic, weak nuclear, and strong
nuclear forces
5B describe and calculate how the magnitude of
the gravitational force between two objects
depends on their masses and the distance
between their centers
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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.
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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
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Circular Motion and Gravitation
Law of Universal Gravitation
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Section 2
Circular Motion and Gravitation
Section 2
The Cavendish Experiment
• Cavendish found the value for G.
– He used an apparatus similar to that shown above.
– He measured the masses of the spheres (m1 and m2), the
distance between the spheres (r), and the force of attraction (Fg).
• He solved Newton’s equation for G and substituted his
experimental values.
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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.
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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?
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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.
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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.
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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
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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
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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?
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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.
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Circular Motion and Gravitation
TEKS
Section 3
The student is expected to:
3D explain the impacts of the scientific
contributions of a variety of historical and
contemporary scientists on scientific thought
and society
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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?
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Circular Motion and Gravitation
Section 3
Kepler’s Laws
• Johannes Kepler built his ideas on planetary motion
using the work of others before him.
– Nicolaus Copernicus and Tycho Brahe
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Circular Motion and Gravitation
Kepler’s Laws
• Kepler’s first law
– Orbits are elliptical, not circular.
– Some orbits are only slightly elliptical.
• Kepler’s second law
– Equal areas are swept out in equal time intervals.
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Section 3
Circular Motion and Gravitation
Section 3
Kepler’s Laws
• Kepler’s third law
– Relates orbital period (T) to distance from the sun (r)
• Period is the time required for one revolution.
– As distance increases, the period increases.
• Not a direct proportion
• T2/r3 has the same value for any object orbiting the sun
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Circular Motion and Gravitation
Section 3
Equations for Planetary Motion
• Using SI units, prove that the units are consistent for
each equation shown above.
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Circular Motion and Gravitation
Section 3
Classroom Practice Problems
• A large planet orbiting a distant star is
discovered. The planet’s orbit is nearly circular
and close to the star. The orbital distance is
7.50  1010 m and its period is 105.5 days.
Calculate the mass of the star.
– Answer: 3.00  1030 kg
• What is the velocity of this planet as it orbits the
star?
– Answer: 5.17  104 m/s
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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.
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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.
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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.
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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.
© Houghton Mifflin Harcourt Publishing Company
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?
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Circular Motion and Gravitation
Section 4
What do you think?
• Doorknobs come in a variety of styles. Describe
some that you have seen.
• Which style of doorknob is easiest to use? Why?
• List the names of any simple machines you can
recall.
• What is the purpose of a simple machine?
• Provide an example.
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Circular Motion and Gravitation
Section 4
Rotational and Translational Motion
• Consider a tire on a moving car.
– Translational motion is the movement of the center of
mass.
• The entire tire is changing positions.
– Rotational motion is the movement around an axis.
• Rotation occurs around a center.
• Changes in rotational motion are caused by
torques.
– Torque is the ability of a force to affect rotation.
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Circular Motion and Gravitation
Section 4
Torque
• Where should the cat push on
the cat-flap door in order to
open it most easily?
– The bottom, as far away from the
hinges as possible
• Torque depends on the force
(F) and the length of the lever
arm (d).
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Circular Motion and Gravitation
Section 4
Torque
• Torque also depends on the angle between the force (F)
and the distance (d).
• Which situation shown above will produce the most
torque on the cat-flap door? Why?
– Figure (a), because the force is perpendicular to the distance
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Circular Motion and Gravitation
Section 4
Torque
• SI units: N•m
– Not joules because torque is not
energy
• The quantity “d sin ” is the
perpendicular distance from the
axis to the direction of the force.
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Circular Motion and Gravitation
Torque as a Vector
• Torque has direction.
– Torque is positive if it causes a
counterclockwise rotation.
– Torque is negative if it causes a
clockwise rotation.
• Are the torques shown to the
right positive or negative?
– The wrench produces a positive
torque.
– The cat produces a negative
torque.
• Net torque is the sum of the
torques.
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Section 4
Circular Motion and Gravitation
Classroom Practice Problems
• Suppose the force on the wrench is
65.0 N and the lever arm is 20.0 cm.
The angle () between the
force and lever arm is 35.0°.
Calculate the torque.
– Answer: 7.46 N•m
• What force would be required to
produce the same torque if the force
was perpendicular to the lever arm?
– Answer: 37.3 N
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Section 4
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
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Circular Motion and Gravitation
Overview of Simple Machines
Click below to watch the Visual Concept.
Visual Concept
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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.
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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
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Circular Motion and Gravitation
Section 4
Now what do you think?
• Doorknobs come in a variety of styles. Describe
some that you have seen.
• Which style of doorknob is easiest to use? Why?
• List the names of any simple machines you can
recall.
• What is the purpose of a simple machine?
• Provide an example.
© Houghton Mifflin Harcourt Publishing Company