Physics 1321 Chapter 13 Lecture Slides.pdf

PowerPoint® Lectures for
University Physics, 14th Edition
– Hugh D. Young and Roger A. Freedman Lectures by Jason Harlow
Gravitation
Chapter 13
© 2016 Pearson Education Inc.

Learning Goals for Chapter 13
Looking forward at …
• how to calculate the gravitational forces that any two bodies
exert on each other.
• how to relate the weight of an object to the general
expression for gravitational force.
• how to calculate the speed, orbital period, and mechanical
energy of a satellite in a circular orbit.
• how to apply and interpret Kepler’s three laws that describe
the motion of planets.
• what black holes are, how to calculate their properties, and
how astronomers discover them.

Introduction
• What can we say
about the motion
of the particles that
make up Saturn’s
rings?
• Why doesn’t the
moon fall to earth,
or the earth into
the sun?
• By studying gravitation and celestial mechanics, we will be
able to answer these and other questions.

Newton’s law of gravitation

Newton’s law of gravitation
• Law of gravitation: Every particle of matter attracts every
other particle with a force that is directly proportional to the
product of their masses and inversely proportional to the
square of the distance between them:
• The gravitational constant G is a fundamental physical
constant that has the same value for any two particles.
• G = 6.67 × 10−11 N · m2/kg2.

Gravitation and spherically symmetric bodies
• The gravitational effect
outside any spherically
symmetric mass distribution
is the same as though all of
the mass were concentrated
at its center.
• The force Fg attracting m1
and m2 on the left is equal to
the force Fg attracting the
two point particles m1 and m2
on the right, which have the
same masses and whose
centers are separated by the
same distance.

Gravitational attraction
• Our solar system is part of a spiral galaxy like this one, which
contains roughly 1011 stars as well as gas, dust, and other
matter.
• The entire assemblage is held
together by the mutual
gravitational attraction of all
the matter in the galaxy.

Weight
• The weight of a body is the total gravitational force exerted
on it by all other bodies in the universe.
• At the surface of the earth, we can neglect all other
gravitational forces, so a body’s weight is:
• The acceleration due to gravity at the earth’s surface is:

Walking and running on the moon
• You automatically transition
from a walk to a run when
the vertical force the ground
exerts on you exceeds your
weight.
• This transition from walking
to running happens at much
lower speeds on the moon,
where objects weigh only
17% as much as on earth.
• Hence, the Apollo astronauts found themselves running even
when moving relatively slowly during their moon “walks.”

Weight decreases with altitude

Interior of the earth
• The earth is approximately
spherically symmetric, but it is
not uniform throughout its
volume.
• The density ρ decreases with
increasing distance r from the
center.

Gravitational potential energy
• The change in gravitational
potential energy is defined as
−1 times the work done by the
gravitational force as the body
moves from r1 to r2.

Gravitational potential energy
• We define the gravitational potential energy U so that
Wgrav = U1 − U2:
• If the earth’s gravitational force on a body is the only force
that does work, then the total mechanical energy of the
system of the earth and body is constant, or conserved.

Gravitational potential energy depends on
distance
• The gravitational potential
energy of the earth–astronaut
system increases (becomes
less negative) as the
astronaut moves away from
the earth.

The motion of satellites
• The trajectory of a
projectile fired from a
great height (ignoring
air resistance) depends
on its initial speed.

Circular satellite orbits
• With a mass of approximately 4.5 × 105 kg and a width of
over 108 m, the International Space Station is the largest
satellite ever placed in orbit.

• For a circular orbit, the
speed of a satellite is just
right to keep its distance
from the center of the earth
constant.
• The force due to the
earth’s gravitational
attraction provides the
centripetal acceleration that
keeps a satellite in orbit.

• A satellite is constantly falling around the earth.
• Astronauts inside the satellite in orbit are in a state of
apparent weightlessness because they are falling with the
satellite.

Kepler’s first law
• Each planet moves in
an elliptical orbit with
the sun at one focus of
the ellipse.

Kepler’s second law
• A line from the sun to a given planet sweeps out equal areas
in equal times.

Kepler’s second law
• Because the gravitational
force that the sun exerts
on a planet produces zero
torque around the sun, the
planet’s angular
momentum around the
sun remains constant.

Kepler’s third law
• The periods of the planets are proportional to the three-halves
powers of the major axis lengths of their orbits.
• Note that the period does not depend on the eccentricity e.
• An asteroid in an elongated elliptical orbit with semi-major
axis a will have the same orbital period as a planet in a
circular orbit of radius a.

Comet Halley
• At the heart of Comet Halley is an icy body, called the
nucleus, that is about 10 km across.
• When the comet’s orbit carries it close to the sun, the heat of
sunlight causes the nucleus to partially evaporate.
• The evaporated material forms the tail, which can be tens of
millions of kilometers long.

Planetary motions and the center of mass
• We have assumed that
as a planet or comet
orbits the sun, the sun
remains absolutely
stationary.
• In fact, both the sun
and the planet orbit
around their common
center of mass.

Spherical mass distributions
• Follow the proof that the
gravitational interaction
between two spherically
symmetric mass
distributions is the same as
if each one were
concentrated at its center.
• To begin, we consider the
gravitational force on a
point mass m outside a
spherical shell.

A point mass inside a spherical shell
• If a point mass is inside a spherically symmetric shell, the
potential energy of the system is constant.
• This means that the shell exerts no force on a point mass
inside of it.
• Only the mass inside
a sphere of radius r
exerts a net
gravitational force
on it.

Apparent weight and the earth’s rotation

Variations of g with latitude and elevation

Black holes
• If a spherical nonrotating body has radius less than the
Schwarzschild radius, nothing can escape from it.
• Such a body is a black hole.
• The surface of the sphere with radius RS surrounding a black
hole is called the event horizon.
• Since light can’t escape from within that sphere, we can’t see
events occurring inside.

Black holes

Detecting black holes
• We can detect black holes by looking for x rays emitted from
their accretion disks.

Physics 1321 Chapter 13 Lecture Slides.pdf

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