AP Physics Chapter 7 Circular Motion and Gravitation

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AP Physics Chapter 7Circular Motion
and Gravitation
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Chapter 7 Circular Motion and Gravitation

• 7.1, 7.2, 7.4 omitted
• 7.3 Uniform Circular Motion and Centripetal
  Acceleration
• 7.5 Newtons Law of Gravitation
• 7.6 Satellites in Circular Orbits (Keplers Laws
  omitted)

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Homework for Chapter 7

• Read Chapter 7
• HW 7.A p.246-247 44-52.
• HW 7.B p.248-249 72-76 78,80.
• HW 7.C see end of packet.

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7.3 Uniform Circular Motion andCentripetal
Acceleration
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Physics Warmup 35
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Physics Warmup 35
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7.3 Uniform Circular Motion and Centripetal
Acceleration
uniform circular motion An object moves at a
constant speed in a circular path.
The speed of an object in uniform circular motion
is constant, but the objects velocity changes in
the direction of motion. Therefore, there is an
acceleration.
Fig. 7.8 p.218
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7.3 Uniform Circular Motion and Centripetal
Acceleration
centripetal acceleration center-seeking For
and object in uniform circular motion, the
centripetal acceleration is directed towards the
center. There is no acceleration component in
the tangential direction. If there were, the
magnitude of the velocity (tangential speed)
would change. ac v2 r
Fc
r
Fig. 7.10, p.219
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7.3 Uniform Circular Motion and Centripetal
Acceleration
From Newtons second law, Fnet ma. Therefore,
there must be a net force associated with
centripetal acceleration. In the case of
uniform circular motion, this force is called
centripetal force. It is always directed toward
the center of the circle since we know the net
force on an object is in the same direction as
acceleration. Fc mac mv2
r Centripetal force is not a
separate or extra force. It is a net force toward
the center of the circle. A centripetal force
is always required for objects to stay in a
circular path. Without it, an object will fly out
along a tangent line due to inertia.
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7.3 Uniform Circular Motion and Centripetal
Acceleration
The time period T, the frequency of rotation f,
the radius of the circular path, and the speed of
the particle undergoing uniform circular motion
are related by T 2 p r 1 v
f centrifugal force center-fleeing force
a fictitious force something made up by
nonphysicists the vector equivalent of a
unicorn Hint Do not label a force as
centripetal force on your free-body diagram
even if that force does act toward the center of
the circle. Rather, label the actual source of
the force i.e., tension, friction, weight,
electric force, etc. Question 1 What provides
the centripetal force when clothes move around a
dryer? (the inside of the dryer) Question 2
What provides the centripetal force upon a
satellite orbiting the Earth?
(Earths gravity)
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Example 7.a
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Example 7.a
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7.3 Uniform Circular Motion and Centripetal
Acceleration

• Example 7.3 A car of mass 1500 kg is negotiating
  a flat circular curve of radius 50 meters with a
  speed of 20 m/s.
• What is the source of centripetal force on the
  car?
• What is the magnitude of the centripetal
  acceleration of the car?
• What is the magnitude of the centripetal force on
  the car?

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7.3 Uniform Circular Motion and Centripetal
Acceleration
Example 7.b
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Example 7.b
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7.3 Uniform Circular Motion and Centripetal
Acceleration
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Check for Understanding 1. In uniform circular
motion, there is a a. constant velocity b.
constant angular velocity c. zero
acceleration d. net tangential acceleration
Answer b
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Check for Understanding 2. If the centripetal
force on a particle in uniform circular motion is
increased, a. the tangential speed will
remain constant b. the tangential speed will
decrease c. the radius of the circular path
will increase d. the tangential speed will
increase and/or the radius will decrease
Answer d
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Check for Understanding 3. Explain why mud
flies off a fast-spinning wheel. Answer
Centripetal force is proportional to the square
of the speed. When there is insufficient
centripetal force (provided by friction and
adhesive forces), the mud cannot maintain the
circular path and it flies off along a tangent.
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Homework 7.A Section 7.3

• HW 7.A p.246-247 44-52.

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7.5 Newtons Law of Gravitation
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Physics Warmup 48
Solution It would decrease. You would have mass
below you pulling downward and mass above you
pulling upward. At the center of the earth, you
would weigh zero.
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7.5 Newtons Law of Gravitation
G is the universal gravitational constant.
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7.5 Newtons Law of Gravitation
F a 1 Inverse Square Law r2

• Gravitational force is the weakest of the four
  fundamental forces.

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For homogeneous spheres, the masses may be
considered to be concentrated at their centers.
Any two particles, or point masses, are
gravitationally attracted to each other with a
force that has a magnitude given by Newtons
universal law of gravitation.
Fig. 7.17, p.228
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7.5 Newtons Law of Gravitation
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7.5 Newtons Law of Gravitation
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7.5 Newtons Law of Gravitation
We can find the acceleration due to gravity,
ag, by setting Newtons 2nd Law the Law of
Gravitation F mag GmM (m cancels
out) r2 so, ag GM This
is the acceleration due to gravity at a
r2 distance r from a planets center.
At the Earths surface agE g GME ME 6.0 x
1024 kg RE2 RE 6.4 x 106
m where ME and RE are the mass and radius of
the Earth. At an altitude h above the Earths
surface ag GME (RE
h)2
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7.5 Newtons Law of Gravitation
Example 7.7 Calculate the acceleration due to
gravity at the surface of the moon. The radius of
the moon is 1750 km and the mass of the moon is
7.4 x 1022 kg.
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7.5 Newtons Law of Gravitation
Note it is just r, not r2, in the denominator.
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7.5 Newtons Law of Gravitation
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Gravitational potential energy
U - Gm1m2 r
Note U mgh only applies to objects near the
surface of the earth.
Fig. 7.20, p. 231
On Earth, we are in a negative gravitational
potential energy well. Work must be done against
gravity to get higher in the well in other
words, U becomes less negative. The top of the
well is at infinity, where the gravitational
potential energy is chosen to be zero.
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7.5 Newtons Law of Gravitation

• Example 7.6 The hydrogen atom consists of a
  proton of mass 1.67 x 10-27 kg and an orbiting
  electron of mass 9.11 x 10-31 kg. In one of its
  orbits, the electron is 5.3 x 10-11 m from the
  proton and in another orbit, it is 10.6 x 10-11 m
  from the proton.
• What are the mutual attractive forces when the
  electron is in these orbits, respectively?
• b) If the electron jumps from the large orbit to
  the small one, what is the change in potential
  energy?

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7.5 Newtons Law of Gravitation
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7.5 Newtons Law of Gravitation Check for
Understanding

• The gravitational force is
• a linear function of distance
• an infinite-range force
• applicable only to our solar system
• sometimes repulsive

Answer b
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7.5 Newtons Law of Gravitation Check for
Understanding

• 2. The acceleration due to gravity on the Earths
  surface
• is a universal constant like G
• does not depend on the Earths mass
• is directly proportional to the Earths radius
• does not depend on the objects mass

Answer d
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7.5 Newtons Law of Gravitation Check for
Understanding
3. Astronauts in a spacecraft orbiting the Earth
or out for a spacewalk are seen to float in
midair. This is sometimes referred to as
weightlessness or zero gravity (zero g). Are
these terms correct? Explain why an astronaut
appears to float in or near an orbiting
spacecraft.
Answer No. Gravity acts on the astronauts and
the spacecraft, providing the necessary
centripetal force for the orbit, so g is not zero
and there is weight by definition (wmg). The
floating occurs because the spacecraft and
astronauts are falling (accelerating toward
Earth at the same rate).
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Homework 7BSection 7.5

• HW 7.B p.248-249 72-76 78,80.

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7.6 Satellites in Circular Orbits
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7.3 Uniform Circular Motion and Centripetal
Acceleration
Physics Warmup 33
Boeing 747
Freedom 7
Space Shuttle
ISS
Hubble
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7.6 Satellites Know How to Derive Both
Formulas
We can find the tangential velocity of a planet
or satellite where m is orbiting
M. Set Centripetal Force Force of
Gravity F mv2 GmM
r r2 Solve for v
v GM tangential velocity
r of an orbiting body A satellites
period can be derived from this expression. Since
v 2 ?? r / T (circumference / period), and M is
the mass of the Sun, 2 ?? r GM
T r Squaring both sides and solving for
T2 gives T2 4 ??2 r3 or T2
Kr3 period of an orbiting body
GM For our solar system, K 2.97 x 10-19
s2/m3
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7.6 Satellites

• Example 11 A satellite is placed into a circular
  orbit 1000 km above the surface of the earth (r
  1000 km 6400 km 7400 km). Determine
• the time period (T) of the satellite
• the speed (v) of the satellite

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7.6 Satellites
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7.6 Satellites
Note a geosynchronous satellite orbits the earth
with a period of 24 hours so its motion is
synchronized with the earths rotation. Viewed by
an observer on earth, a geosynchronous satellite
appears to be stationary. All geosynchronous
satellites with circular orbits have the same
orbital radius (36,000 km or 22,000 mi above sea
level for Earth).
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7.6 Check for Understanding
A Space Shuttle orbits Earth 300 km above the
surface. Why cant the Shuttle orbit 10 km above
Earth? a) The Space Shuttle cannot go fast
enough to maintain such an orbit. b) Because r
appears in the denominator of Newtons law of
gravitation, the force of gravity is much larger
closer to the Earth this force is too strong to
allow such an orbit. c) The closer orbit would
likely crash into a large mountain such as
Everest because of its elliptical nature. d)
Much of the Shuttles kinetic energy would be
dissipated as heat in the atmosphere, degrading
the orbit.
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7.6 Check for Understanding
Answer d. A circular orbit is allowed at any
distance from a planet, as long as the satellite
moves fast enough. At 300 km above the surface
Earths atmosphere is practically nonexistent. At
10 km, though, the atmospheric friction would
quickly cause the shuttle to slow down.
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7.6 Satellites
7.6 Check for Understanding
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7.6 Satellites
7.6 Check for Understanding
The period of a satellite is given by the
formula T2 K r3. This means a specific period
maps onto a specific orbital radius. Therefore,
there is only one orbital radius for a
geosynchronous satellite with a circular
orbit.
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7.6 Check for Understanding
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7.6 Check for Understanding
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Internet Activity Put a satellite in
orbit http//www.lon-capa.org/mmp/kap7/orbiter/
orbit.htm
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Homework 7.C Section 7.6

• Problems
• 1. For a geosynchronous communications satellite
  (time period 24 hours) determine,
• a) the distance from the surface of the earth
• b) the speed of the satellite
• 2. The moons orbit around the earth is
  approximately a circle of radius 3.84 x 108 m.
  Its period is about 27.3 days. With this
  information determine the mass of the earth.
• 3. Calculate the mass of the planet Jupiter from
  the following information
• Io, one of the moons of Jupiter has a circular
  orbit of radius 4.2 x 108 m and its period is
  42.5 hours.
• 4. An object at rest on the surface of the earth
  completes one orbit around the earth in 24
  hours. But this object is not a satellite. What
  would be the time period of a hypothetical
  satellite at the level of the earths surface?

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Homework 7.C Section 7.6

• Multiple Choice
• 5. A satellite A of mass m and a satellite B of
  mass 2m are both in the same circular orbit
  around the earth. Which of the following is true
  about these satellites?
• I. Both have same speed
• II. Both have same acceleration
• III. The KE of B is twice that of A
• a) all three b) I and II only c) III only
• d) I and III only e) none of these
• 6. Time period of a satellite in circular orbit
  around a planet does not depend on
• a) its distance from the planet b) mass of the
  satellite
• c) gravitational field of the planet d) mass of
  the planet
• e) all of these
• 7. Which of the following is responsible for
  astronauts in orbit around the earth feeling
  weightless?
• a) very weak or no gravity at that distance
• b) they are in a vacuum
• c) there is no normal force supporting them as
  they are in free fall
• d) the pull of the earth on them is cancelled by
  the pull of the moon

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• Warmup TSAR
• Think (3 minutes)
• quietly about the following question
• How does Newtons Law of Gravitation relate to
  the motion of satellites?
• Write your answer using full sentences in
  paragraph form. You may include formulas and
  sketches.
• Share (3 minutes)
• Pair up. Read your paragraph word for word to
  your partner. Explain any sketches.
• Your partner will listen silently and prepare
  to give quality feedback. Your partner may not
  give feedback yet, but they may ask for
  clarification.
• Now, switch roles.

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• Advise (3 minutes)
• Your partner will now give you advice on your
  answer. Listen to your partner, ask questions to
  clarify, and evaluate the advice.
• Now, switch roles.
• Revise (3 minutes)
• Decide what advice was useful.
• Revise work in red pen.

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End of Chapter 7