Transcript 7.3 Uniform Circular Motion and Centripetal

Lecture Outline
Chapter 7
College Physics, 7th Edition
Wilson / Buffa / Lou
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Chapter 7
Circular Motion and
Gravitation
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Units of Chapter 7
Angular Measure
Angular Speed and Velocity
Uniform Circular Motion and Centripetal
Acceleration
Angular Acceleration
Newton’s Law of Gravitation
Kepler’s Laws and Earth Satellites
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7.1 Angular Measure
The position of an
object can be described
using polar
coordinates—r and θ—
rather than x and y. The
figure at left gives the
conversion between the
two descriptions.
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7.1 Angular Measure
• r is a distance that extends from the origin.
r is the same for any point on a given
circle. (like the radius!)
• Θ is an angle, and it changes with time.
• Linear Displacement…how do we
calculate?
• Angular Displacement is VERY similar
7.1 Angular Measure
• Δθ = θ - θi
• The unit for angular displacement is the
degree.
• There are 360 degrees in one complete
circle.
7.1 Angular Measure
• The arc length, s, is
the distance that is
traveled along the
circular path.
• The θ is said to define
the arc length.
• It is most convenient
to measure the angle
θ in radians.
7.1 Angular Measure
Relationship between arc length, the
radius, and the angle:
For one full circle,
with s = 2πr
(this is the
circumference of
the circle)
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7.1 Angular Measure
• A spectator standing at the
center of a circular running
track observes a runner start a
practice race 256m due east of
her own position. The runner
runs on the track to the finish
line, which is located due north
of the observer’s position.
What is the distance of the
run?
7.1 Angular Measure
• A sailor sights a distance
tanker ship and finds that
it subtends an angle of
1.15 degrees. He knows
from the shipping charts
that the tanker is 150m in
length. Approximately
how far away is the
tanker?
7.2 Angular Speed and Velocity
• How do we calculate speed?
• What’s the difference between average
speed and instantaneous speed?
7.2 Angular Speed and Velocity
In analogy to the linear case, we define the
average and instantaneous angular speed:
Units??
Angular Velocity??
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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.
How does this work?
Counterclockwise is
positive
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7.2 Angular Speed and Velocity
• A particle moving in a circle has an
instantaneous velocity tangential to its
circular path.
• What is a tangent?
• Tangential speed (the particle’s orbital
speed)
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.
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7.2 Angular Speed and Velocity
• An amusement park merry go
round at its constant
operational speed makes one
complete rotation in 45
seconds. Two children are on
horses, one at 3.0 m from the
center of the ride and the other
farther out at 6.0 m from the
center.
– What are the angular speeds
of each?
– What are the tangential
speeds of each?
7.2 Angular Speed and Velocity
The period is the time it takes for one complete
revolution (rotation)
For example: The period of revolution of the
Earth around the Sun is one year.
Or the period of the Earth’s axial rotation is 24
hours.
Units: seconds or sometimes seconds/cycle
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7.2 Angular Speed and Velocity
• the frequency is the number of revolutions
(rotations) per second. [Units: Hertz]
• The relation of the frequency to the
angular speed:
7.2 Angular Speed and Velocity
• A CD rotates in a player at a constant
speed of 200 rpm. What are the CD’s
– Frequency?
– Period?
7.3 Uniform Circular Motion and
Centripetal Acceleration
• What is uniform
motion?
• What is uniform
circular motion?
7.3 Uniform Circular Motion and
Centripetal Acceleration
• The acceleration in uniform circular motion
is called centripetal acceleration.
• Centripetal means “center-seeking.”
• Centripetal acceleration is directed inward
or “into” the circle.
• The tangential velocity is perpendicular to
the centripetal acceleration.
7.3 Uniform Circular Motion and
Centripetal Acceleration
Instantaneous centripetal
acceleration
Can also be written as…
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7.3 Uniform Circular Motion and
Centripetal Acceleration
• A laboratory centrifuge operates at a
rotational speed of 12,000 rpm.
– What is the magnitude of the centripetal
acceleration of a red blood cell at a radial
distance of 8.00 cm from the centrifuge’s axis
of rotation?
– How does this acceleration compare with g?
7.3 Uniform Circular Motion and
Centripetal Acceleration
The centripetal force (net inward 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.
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7.3 Uniform Circular Motion and
Centripetal Acceleration
• A ball is attached to a
string is swung with
uniform motion in a
horizontal circle
above a person’s
head. If the string
breaks, which of the
trajectories shown on
the following slide
would the ball follow.
7.4 Angular Acceleration
The average angular acceleration is the rate at
which the angular speed changes:
In analogy to constant linear acceleration:
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7.4 Angular Acceleration
If the angular speed is
changing, the linear
speed must be
changing as well. The
tangential acceleration
is related to the angular
acceleration:
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7.4 Angular Acceleration
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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:
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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:
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7.5 Newton’s Law of Gravitation
The gravitational potential energy is given by
the general expression:
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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.
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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.
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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.
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7.6 Kepler’s Laws and Earth Satellites
If a projectile is given enough speed to just
reach the top of the Earth’s gravitational well,
its potential energy at the top will be zero. At
the minimum, its kinetic energy will be zero
there as well.
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7.6 Kepler’s Laws and Earth Satellites
This minimum initial speed is called the
escape speed.
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7.6 Kepler’s Laws and Earth Satellites
Any satellite in orbit around the
Earth has a speed given by
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7.6 Kepler’s Laws and Earth Satellites
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7.6 Kepler’s Laws and Earth Satellites
Astronauts in Earth orbit report the sensation
of weightlessness. The gravitational force on
them is not zero; what’s happening?
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7.6 Kepler’s Laws and Earth Satellites
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.
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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:
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Summary of Chapter 7
Tangential speed is proportional to angular
speed.
Frequency is inversely proportional to period.
Angular speed:
Centripetal acceleration:
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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:
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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.
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Summary of Chapter 7
Escape speed from Earth:
Energy of a satellite orbiting Earth:
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