Transcript WBL6_Lecture_Ch07

Lecture Outlines
Chapter 7
College Physics, 6th Edition
Wilson / Buffa / Lou
© 2007 Pearson Prentice Hall
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Chapter 7
Circular Motion and
Gravitation
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
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.
7.1 Angular Measure
It is most convenient to measure the
angle θ in radians:
7.1 Angular Measure
The small-angle
approximation is very
useful, as it allows the
substitution of θ for sin θ
when the angle is
sufficiently small.
7.2 Angular Speed and Velocity
In analogy to the linear case, we define the
average and instantaneous angular speed:
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.
The relation of the frequency to the angular
speed:
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.
7.4 Angular Acceleration
The average angular acceleration is the rate at
which the angular speed changes:
In analogy to constant linear acceleration:
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:
7.4 Angular Acceleration
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 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:
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.
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.
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.
7.6 Kepler’s Laws and Earth Satellites
This minimum initial speed 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
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?
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.
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:
Energy of a satellite orbiting Earth: