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Chapter 7
Circular Motion and
Gravitation
7.1 Angular Measure
The position of an object can
be described using polar
coordinates—r and θ—rather
than x and y.
7.1 Angular Measure
The distance r extends from
the origin, and the angle Θ is
commonly measured counterclockwise from the positive xaxis
7.1 Angular Measure
The length of the arc between
r and the x-axis is known as
the ―arclength‖ (see what they did there?)
and is represented by the
variable ‗s‘
An object ―subtends‖ an
angle. In other words… the
angle that the object takes up.
7.1 Angular Measure
It is most convenient to measure the
angle θ in radians:
7.1 Angular Measure
It is extremely handy to know that…
1 rad = 57.3°
360° = 2π rad
7.1 Angular Measure
1. What is the radius of the Earth if walking
around it twice is a distance of 8.02E4 km?
2. The moon is 4.08E8 m away from Earth.
If it moves an angle of 1 rad through the sky,
what distance has it actually moved?
7.2 Angular Speed and Velocity
In analogy to the linear case, we define the
average and instantaneous angular velocity:
Δd
v=
Δt
d
v=
t
7.2 Angular Speed and Velocity
The direction of the
angular velocity is along
the axis of rotation, and is
given by a right-hand rule.
7.2 Angular Speed and Velocity
Relationship between tangential and angular
speeds:
This means that parts
of a rotating object
farther from the axis of
rotation move faster.
7.2 Angular Speed and Velocity
The period is the time it takes for one rotation;
the frequency is the number of rotations per
second.
f = frequency
T = period
The relation of the frequency to the angular
velocity:
Warm-Up Questions
1.
Fred runs four and a half laps around a circular track. If this took
him seven minutes at a constant speed, what is his angular
displacement and angular velocity?
2.
Bob is swinging his 0.30 meter long ID lanyard around at 2 rev/s
when he lets go, what is the tangential velocity of the ID right
when it is let go?
3.
The moon is 3.84x105 km from the Earth. If the distance from the
space station to the Earth‘s center is is 6.4x103 km, how many
orbits must they complete to have travelled a distance equal to
that of a journey to the moon?
4.
From #3, if it takes them 90 minutes to orbit the earth what is their
angular velocity?
Recap from Yesterday!
r = meters
Θ = radians
Θ = radians
t = seconds
r = meters
ω = rad/s
s = … meters?!
ω = rad/s
v = m/s
(why?)
7.3 Buckets of Fun
What is happening?
Does the water fall?
What direction is the
overall acceleration?
What velocity or velocities
does the bucket have?
7.3 Uniform Circular Motion and
Centripetal Acceleration
A careful look at the change in the velocity
vector of an object moving in a circle at
constant speed shows that the acceleration is
toward the center of the circle.
7.3 Uniform Circular Motion and
Centripetal Acceleration
The same analysis
shows that the
centripetal acceleration
is given by:
7.3 Uniform Circular Motion and
Centripetal Acceleration
The centripetal force is the mass multiplied by
the centripetal acceleration.
This force is the net force on the object. As
the force is always perpendicular to the
velocity, it does no work.
Centrifuge
Centripetal? Centrifugal?
•
The centripetal force constantly accelerates
the object towards the center of rotation.
•
The centrifugal force has two potential
definitions (one real, one not so real).
•
The misunderstood idea of centrifugal force
is that it is a force pulling the rotating object
outward. – There is no such force!
•
In reality, the centrifugal force is the reactive
force of the object pushing against whatever
is making it rotate.
– Water against the bucket
– The object wants to keep going with the
tangential velocity, but its spinning
container prevents this.
• 1. The largest salami in the world, made in Norway, was more than
20 m long. If a hungry mouse ran around the salami‘s circumference
with a tangential speed of 0.17 m/s, the centripetal acceleration of
the mouse was 0.29 m/s2.What was the radius of the salami?
• 2. The largest steerable single-dish radio telescope is located at the
branch of the Max Planck Institute in Bonn, Germany. Suppose this
telescope rotates about its axis with the same angular speed as
Earth. The centripetal acceleration of the points at the edge of the
telescope is 2.65 x 10-7m/s2. What is the radius of the telescope
dish?
• 3. In 1994, Susan Williams of California blew a bubble-gum bubble
with a diameter of 58.4 cm. If this bubble was rigid and the
centripetal acceleration of the equatorial points of the bubble were
8.50 x 102 m/s2, what would the tangential speed of those points be?
• 4. An ostrich lays the largest bird egg. A typical diameter for an
ostrich egg at its widest part is 12 cm. Suppose an egg of this size
rolls down a slope so that the centripetal acceleration of the shell at
its widest part is 0.28 m/s2.What is the tangential speed of that part
of the shell?
• 5. A waterwheel built in Hamah, Syria, rotates continuously. The
wheel‘s radius is 20.0 m. If the wheel rotates once in 16.0 s, what is
the magnitude of the centripetal acceleration of the wheel‘s edge?
• 6. In 1995, Cathy Marsal of France cycled 47.112 km in 1.000 hour.
Calculate the magnitude of the centripetal acceleration of Marsal
with respect to Earth‘s center. Neglect Earth‘s rotation, and use 6.37
x 103 km as Earth‘s radius.
7.4 Angular Acceleration
The average angular acceleration is the rate at
which the angular speed changes:
If the angular speed is
changing, the linear speed
must be changing as well.
Linear Units Recap
Linear...
Variable
Symbol
Unit
Displacement
Velocity
Acceleration
Mass
Force
Momentum
x
v
a
m
F
p
m
m/s
2
m/s
kg
N
kg·m/s
Angular Units Recap
Angular...
Variable
Symbol
Unit
Displacement
Velocity
Acceleration
Mass
Force
Momentum
θ
ω
α
??
??
??
rad
rad/s
2
rad/s
??
??
??
7.4 Angular Acceleration
Ch. 7 Equations
1. Peter Rosendahl of Sweden rode a unicycle with a wheel
diameter of 2.5 cm. If the wheel‘s average angular acceleration
was 2.0 rad/s2, how long would it take for the wheel‘s angular
speed to increase from 0 rad/s to 9.4 rad/s?
2. Jupiter has the shortest day of all of the solar system‘s
planets. One rotation of Jupiter occurs in 9.83 h. If an average
angular acceleration of –3.0 × 10–8 rad/s2 slows Jupiter‘s
rotation, how long does it take for Jupiter to stop rotating?
3. In 1989, Dave Moore of California built the Frankencycle, a
bicycle with a wheel diameter of more than 3 m. If you ride this
bike so that the wheels‘ angular speed increases from 2.00
rad/s to 3.15 rad/s in 3.6 s, what is the average angular
acceleration of the wheels?
4. In 1990, David Robilliard rode a bicycle on the back wheel for
more than 5 h. If the wheel‘s initial angular speed was 8.0 rad/s and
Robilliard tripled this speed in 25 s, what was the average angular
acceleration?
5. Earth takes about 365 days to orbit once around the sun.
Mercury, the innermost planet, takes less than a fourth of this time
to complete one revolution. Suppose some mysterious force causes
Earth to experience an average angular acceleration of 6.05 × 10–13
rad/s2, so that after 12.0 days its angular orbital speed is the same
as that of Mercury. Calculate this angular speed and the period of
one orbit.
6. The smallest ridable tandem bicycle was built in France and has a
length of less than 40 cm. Suppose this bicycle accelerates from
rest so that the average angular acceleration of the wheels is 0.800
rad/s2.What is the angular speed of the wheels after 8.40 s?
Newton‘s Epiphany
7.5 Newton’s Law of Gravitation
Newton’s law of universal gravitation describes
the force between any two point masses:
G is called the universal gravitational
constant:
7.5 Newton’s Genius
It was through logic, reasoning, and luck that
Newton’s work led to this equation and the value
for his constant, G. Using this equation he was
able to calculate the mass
of the Earth, which in turn
allowed the mass of the
moon, sun and other
planets to be determined!
But… how did he do it?
7.5 Newton’s Law of Gravitation
Gravity provides the centripetal force that
keeps planets, moons, and satellites in their
orbits.
We can relate the universal gravitational force
to the local acceleration of gravity:
7.5 Newton’s Law of Gravitation
The gravitational potential energy is given by
the general expression:
Einstein / Newton
Higgs Particle?
• Last week there was a press conference where it was
announced that they have some additional evidence of a
―Higgs‖ Particle
– Predicted to exist mathematically
– May be responsible for giving other particles mass – huge step
towards understanding how gravity actually works (not just
understanding how to calculate it)
1. A giant asteroid flying towards us has an average radius of 4.3 km. If the
gravitational force between the asteroid and a 5.0 kg rock at its surface is
4.5 ×10–2 N, what is the mass of the asteroid?
2. The largest turtle ever found has a mass of 621 kg. If the force of
gravitational attraction between this turtle and a person with a mass of
65.0 kg is 1.0 × 10–12 N, what is the distance between them?
3. A group of ants descended on a picnic basket. The group‘s mass was
estimated to be 3.2 × 109 kg. If this group was split in half and the halves
separated by 1.0 × 102 m, what would the magnitude of the gravitational
force between the halves be?
1. In the year 34928 Earth has won the War of the Universe by obliterating
all of the other planetary bodies in the entire universe, becoming the only
mass. A 5.8x104 kg starship is sent away from Earth, travelling a
distance of 150 light-years away. What is the gravitational potential
energy of the starship?
2. Why can‘t the potential energy ever be zero or positive when considered
this way? (answer is not specific to the math)
1. The radius of the Earth is 6378 km and its mass is 5.97x1024 kg.
Compare the acceleration of gravity for you on Earth to the
acceleration experienced by astronauts in the International Space
Station 322 km above the Earth’s surface.
2. What centripetal acceleration does the ISS experience?
3. What is the angular velocity of the ISS?
4. What is the tangential velocity of the ISS?
5. What is the period of one orbit for the ISS?
“Zero Gravity”
Astronauts in Earth orbit report the sensation
of weightlessness. The gravitational force on
them is not zero; what’s happening?
“Zero Gravity”
What’s missing is not the weight, but the
normal force. We call this apparent
weightlessness.
“Artificial” gravity could be produced in orbit
by rotating the satellite; the centripetal force
would mimic the effects of gravity.
Johannes Kepler
(1571-1630)
Mathematician
Astronomer
Astrologer
Supernova of 1604
Revelation
7.6 Kepler’s Laws and Earth Satellites
Kepler‘s laws were the result of his many years of
observations. They were later found to be
consequences of Newton‘s laws.
Kepler’s first law:
Planets move in elliptical orbits, with the Sun at one of
the focal points.
Earth‘s Orbit
Earth‘s Orbit
• Aphelion:
• Perihelion:
152,091,221 km
147,095,260 km
• Very close to circular orbit
• Also changes on a 100,000 year cycle due
to gravitational influences from
neighboring masses (primarily Jupiter)
7.6 Kepler’s Laws and Earth Satellites
Kepler’s second law:
A line from the Sun to a planet sweeps out equal areas in
equal lengths of time.
7.6 Kepler’s Laws and Earth Satellites
Kepler’s third law:
The square of the orbital period of a planet is directly
proportional to the cube of the average distance of the
planet from the Sun; that is,
.
This can be derived from Newton‘s law of
gravitation, using a circular orbit.
Basically: there exists a relationship between
period and radius of the circular path.
7.6 Kepler’s Laws and Earth Satellites
This minimum initial speed to leave orbit
around a planetary body is called the
escape speed.
7.6 Kepler’s Laws and Earth Satellites
Any satellite in orbit around the
Earth has a speed given by
7.6 Kepler’s Laws and Earth Satellites
Escape Velocity Practice
1.
The mass of the Earth is 5.97  1024 kg and its radius is
6371 km. Find the escape velocity from the earth.
2.
Suppose the Earth shrunk suddenly to one-fourth its
radius without any change in its mass. What would be
the escape velocity then?
3.
An imaginary planet X has a mass eight times that of the
earth and radius twice that of the earth. What would be
the escape velocity from this planet?
4.
Your Physics textbook has a mass of _____, to what
radius would you need to compress it to make a black
hole? (vesc of black hole = speed of light)
Summary of Chapter 7
Angles may be measured in radians; the angle
is the arc length divided by the radius.
Angular kinematic equations for constant
acceleration:
Summary of Chapter 7
Tangential speed is proportional to angular
speed.
Frequency is inversely proportional to period.
Angular speed:
Centripetal acceleration:
Summary of Chapter 7
Centripetal force:
Angular acceleration is the rate at which the
angular speed changes. It is related to the
tangential acceleration.
Newton’s law of gravitation:
Summary of Chapter 7
Gravitational potential energy:
Kepler’s laws:
1. Planetary orbits are ellipses with Sun at one
focus
2. Equal areas are swept out in equal times.
3. The square of the period is proportional to
the cube of the radius.
Summary of Chapter 7
Escape speed from Earth:
Chapter 7 Review
Pages 249-254:
Questions:
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11-15
27-32
44, 46-49, 53
66-67
77-78, 80-82, 84
91-93
7.1: Angular Measure
7.2: Angular Velocity
7.3: Centripetal Acceleration
7.4: Angular Acceleration
7.5: Newton‘s Law of Grav.
7.6: Kepler‘s Laws
Mearth = 5.97x1024 kg
Rearth = 6371 km
Mearth = 5.97x1024 kg
Rearth = 6371 km
Mearth = 5.97x1024 kg
Rearth = 6371 km
Mearth = 5.97x1024 kg
Rearth = 6371 km
Mearth = 5.97x1024 kg
Rearth = 6371 km
Mearth = 5.97x1024 kg
Rearth = 6371 km