Transcript Circular Motion and Gravitation

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
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
Tangential Speed (vt)
• Period= amount of
time to travel around
the circle
• Distance around
circle = circumference
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Section 1
Circular Motion and Gravitation
Centripetal Acceleration (ac)
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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
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
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.
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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.
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