Chapter 13 Gravitation

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Chapter 13 Gravitation

• Newtons Law of Gravitation
• Shell Theorem
• Gravitational Potential Energy In a more
  general way
• Keplers Law
• Orbits and Energy

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13-2 Newtons Law of Universal Gravitation
In 1665 Isaac Newton (23 years old) suggested
that every body in the universe attracts every
other body this tendency of bodies to move
towards each other is called GRAVITATION.
Newton proposed a FORCE LAW
G is called the gravitational constant
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Shell Theorem
Newtons Law as mentioned in the last slide
applies strictly to particles. What about real
objects of finite size? For example how do we
find the gravitational force between an apple and
the earth? Newton solved the apple Earth
problem by proving an important theorem called
shell theorem
A uniform spherical shell of matter attracts a
particle that is outside the shell as if all the
shells mass were concentrated at its center.
Earth can be thought of as a nest of such shells.
Thus from the apples point of view the Earth
does behave like a particle, one that is located
at the center of the Earth and has a mass equal
to that of the Earth.
F ?
F ?
m
M
r
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Falling Around the Earth
x v t
v
y 0.5 g t 2

Newton imagined a cannon ball fired
horizontally
from a mountain top at a speed
v. In a time t it
falls a distance y 0.5 g t 2 while
moving horizontally a
distance x v t. If fired fast
enough (about 8 km/s), the Earth would
curve downward the same
amount the cannon ball falls downward. Thus, the
projectile would never hit the
ground, and it would be in orbit. The
moon falls around Earth in the exact same way
but at a much greater altitude.

.
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13-3 Gravitation and the Principle of
Superposition
Sample Problem 13-1
Figure 13-4a shows an arrangement of three
particles, particle 1 having mass m1 6.0 kg and
particles 2 and 3 having mass m2 m3 4.0 kg,
and with distance a 2.0 cm. What is the net
gravitational force that acts on particle 1
due to the other particles?
The force on particle 1 from the other two
particles is found by adding (vectorially) the
forces particle 2 and 3 would individually exert
on particle 3 if only one of them (2 or 3) is the
other particle present. This is an example of
applying the Principle of Superposition
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Sample Problem 13-2
Figure 14-5a shows an arrangement of five
particles, with masses m1 8.0 kg, m2 m3 m4
m5 2.0 kg, and with a 2.0 cm and q 30.
What is the net gravitational force on
particle 1 due to the other particles?
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13-4 Gravitation Near Earths Surface
Assuming the earth to be a uniform sphere of mass
M, the magnitude of the gravitational force F on
a particle of mass, located outside earth a
distance r from earths center is
If the particle is released it will fall toward
the center of earth, as a result of the
gravitational force F, with an acceleration we
shall call the Gravitational Acceleration, ag.
(also called Gravitational Field, i.e.
gravitational force per unit mass)
Newtons second law
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Earths gravitational field
Gravitational force acts from a distance through
a field
Close to the surface
Far away from the surface
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For objects on the surface of the earth
Is this exactly correct?
The gravitational acceleration,
as calculated above can differ from the actual
measurement of the free fall acceleration g! WHY?
The table in the next page shows values of ag
computed for various altitude above earths
surface. The actual free fall acceleration g of a
particle will differ from ag (calculated) over
the earths surface for three reasons

• Earth is not uniform
• Earth is not a perfect sphere
• Earth rotates

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Earth is not uniform. The density of earth is not
uniform.
The Earth is not a sphere. The Earths radius is
greater at its equator by 21 km than at its
poles.
Compare 12700km (diameter of earth) with 21km
!! Everest mountain is 8 km high!
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The measured weight is the normal force of the
scale.
Earth rotates.
The measured acceleration is less than the
gravitational acceleration, because of Earths
rotation
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13-5 Gravitation Inside Earth

• A uniform shell of matter exerts no net
  gravitational force on a particle located in side
  it.
• The gravitational force at a point r measured
  from the center of Earth comes entirely from the
  mass within the sphere of radius r.

• In Pole to Pole, an early science fiction story
  by George Griffith, three explorers attempt to
  travel by capsule through a naturally formed
  (and, of course, fictional) tunnel directly from
  the south pole to the north pole. According to
  the story, as the capsule approaches Earth's
  center, the gravitational force on the explorers
  becomes alarmingly large and then, exactly at the
  center, it suddenly but only momentarily
  disappears. Then the capsule travels through the
  second half of the tunnel, to the north pole.

Sample Problem 13-4
Check Griffith's description by finding the
gravitational force on the capsule of mass m when
it reaches a distance r from Earth's center.
Assume that Earth is a sphere of uniform density
r (mass per unit volume).
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Sample Problem 13-4 cont..
The gravitational force at distance r from the
center is
The force obeys Hookes Law, which means that the
mass would oscillate like an object on a spring.
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13-6 Gravitational Potential Energy

• Notice the negative (-) sign
• The potential energy approaches zero as r
  approaches infinity.
• The potential energy is negative at a finite
  value of r.

For a system of three particles
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Gravitational Potential Energy of the Earth

• Notice the sign
• U 0 at infinity

• U will get smaller (more negative) as r gets
  smaller.
• Falling down means loosing gravitational
  potential energy.
• Use only when far away from earth otherwise use
  approximation DU mg Dy.

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Potential Energy and Forces
Derive the force from the potential energy
function
The minus sign means that the force on mass m
points radially inward toward mass M.
The Escape Speed
Consider a projectile of mass m, leaving the
surface of a planet with escape speed vesc. It
has kinetic energy K given by and a
potential energy given by
From the principle of conservation of energy Ei
Ef
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Sample Problem 13-5

• An asteroid, headed directly toward Earth, has a
  speed of 12 km/s relative to the planet when it
  is at a distance of 10 Earth radii from Earth's
  center. Neglecting the effects of Earth's
  atmosphere on the asteroid, find the asteroid's
  speed vf when it reaches Earth's surface.

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13-7 Planets and Satellites Keplers Laws

• The Law of Orbits All planets move in elliptical
  orbits, with the Sun at one focus.

• F and F' are the focal points and the sun is at
  one focus F.
• a semimajor axis
• e eccentricity such that ea is the distance
  from the center of the ellipse to either focus F
  or F'
• Ra aphelion farthest from the sun
• Rp perihelion nearest to the sun

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• The Law of Areas A line that connects a planet
  to the Sun sweeps out equal areas in the plane of
  the planets orbit in equal times that is, the
  rate dA/dt at which it sweeps out area A is
  constant.

Area S-A-B equals area S-D-C
Keplers second law is indeed equivalent to the
law of conservation of angular momentum.
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• The Law of Periods The square of the period of
  any planet is proportional to the cube of the
  semimajor axis of the orbit.

This equation holds also for elliptical orbits,
provided we replace r with a, the semimajor axis
of the ellipse. In general one can say
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Keplers laws about planetary motion
Most planets, except Mercury and Pluto, are on
almost a circular orbit
For all planets Mp ltlt Ms
EarthRatio of minor to major axis b/a
0.99986. e 0.017
For planets around sun
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13-8 Satellites Orbits and Energy
Equating the gravitational force and centripetal
force
Noting the potential energy is
we have the relation for a satellite
in a circular orbit as
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Sample Problem 13-8

• A playful astronaut releases a bowling ball, of
  mass m 7.20 kg, into circular orbit about Earth
  at an altitude h of 350 km.
• What is the mechanical energy E of the ball in
  its orbit?

• What is the mechanical energy Eo of the ball on
  the launch pad at Cape Canaveral? From there to
  the orbit, what is the change DE in the ball's
  mechanical energy?

The mechanical energy of the bowling ball at the
surface of the Earth is
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Sample Problem 13-8
Neglecting the Earths rotation, the kinetic
energy of the bowling ball is zero. Therefore,
the M.E. of the ball is
The M.E. difference between at launch pad and
orbit is