Transcript The Law of Universal Gravitation

The Law of Universal
Gravitation
Physics
Montwood High School
R. Casao
Newton’s Universal Law of Gravity
Legend has it that Newton was struck on the head
by a falling apple while napping under a tree.
This prompted Newton to imagine that all bodies
in the universe are attracted to each other in the
same way that the apple was attracted to the
Earth.
Newton analyzed astronomical data on the motion
of the Moon around the Earth and stated that the
law of force governing the motion of the planets
has the same mathematical form as the force law
that attracts the falling apple to the Earth.
Newton’s Universal Law of Gravity
Newton’s law of gravitation: every particle in the
universe 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.
If the particles have masses m1 and m2 and are
separated by a distance r, the magnitude of the
gravitational force is:
Fg 
G  m1  m 2
r
2
Newton’s Universal Law of Gravity
G is the universal gravitational constant, which
has been measured experimentally as 6.672 x
10-11
.
N  m2
kg2
The distance r between m1 and m2 is measured
from the center of m1 to the center of m2.
Newton’s Universal Law of Gravity
By Newton’s third law,
the magnitude of the
force exerted by m1 on
m2 is equal to the force
exerted by m2 on m1,
but opposite in
direction. These
gravitational forces
form an actionreaction pair.
Properties of the Gravitational Force:
The gravitational force acts as an action-ata-distance force, which also exists between
two particles, regardless of the medium
that separates them.
The force varies as the inverse square of the
distance between the particles and
therefore decreases rapidly with increasing
distance between the particles.
The gravitational force is proportional to
the mass of each particle.
Properties of the Gravitational Force:
The force on a particle of mass m at the Earth’s surface
has the magnitude:
G  ME  m
Fg 
2
RE
ME is the Earth’s mass (5.98 x 1024 kg); RE is the radius
of the Earth (6.37 x 106 m).
The net force is directed toward the center of the Earth;
both masses accelerate, but the Earth’s acceleration is
not noticeable due to its extremely large mass. The
smaller mass accelerates towards the Earth.
Weight and Gravitational Force
Weight was previously defined as FW = m·g,
where g is the magnitude of the acceleration due
to gravity. With the new perspective related to
the attractive forces existing between any two
objects in the universe,
G  ME  m
mg 
R E2
The mass m cancels out, giving us (also called
surface gravity):
G  ME
g
2
RE
Bodies Above the Surface of a Mass
Consider a body of mass m at a
distance h above the Earth’s surface, or
a distance r from the Earth’s center,
where r = Re + h. The magnitude of the
gravitational force acting on the mass
is given by:
G  ME  m
Fg 
2
RE  h


Gravity/Radius Ratio
If the body is in free fall, then the acceleration of
gravity at the altitude h is given by:
g
G  ME
R E  h
2
Thus, it follows that g decreases with increasing
altitude.
Gravity/Radius Ratio
The value of g at any given location can be
determined using the following proportional
relationship:
2
g1 r2
 2
g 2 r1
This proportional relationship can also be
applied to the weight of an object:
Fw1
Fw 2

r2
2
2
r1
Kepler’s Laws
Kepler formulated three kinematic laws to
describe the motion of planets about the
Sun:
Kepler’s First Law: All planets move in elliptical
orbits with the sun at one of the focal points.
 Kepler’s Second Law: The radius vector drawn
from the sun to any planet sweeps out equal areas in
equal time intervals.
 Kepler’s Third Law: The square of the orbital
period of any planet is proportional to the cube of
the semi-major axis of the elliptical orbit.

Kepler’s Laws
First law:
Second law:
Kepler’s Laws
Third law equation:



k
r3
T2
where k is a constant 3.35 x 1018 m3/s2
r is the radius of rotation
T is the period of rotation (the time necessary to complete
one revolution)
Kepler’s laws apply to any body that orbits the Sun,
manmade spaceship as well as planets, comets, and
other natural objects. The mass of the orbiting body
does not enter into the calculation.
Kepler’s Laws
The ratio of the squares of the periods (T)
of any two planets revolving about the Sun
is equal to the ratio of the cubes of their
average distances r from the Sun:
2
T1
2
T2

3
r1
3
r2
Period of a Satellite
The period of a satellite or planet orbiting about a
central body is given by:
2
3
4 r
2
T 

G  Mbody
Mbody is the mass of the central body being
orbited.
The Gravitational Field
The gravitational field concept revolves around the
general idea that an object modifies the space
surrounding it by establishing a gravitational field
which extends outward in all directions, falling to zero
at infinity.


Any other mass located within this field experiences a force
because of its location. So, it is the strength of the
gravitational field at that location that produces the force not the distant object.
The situation is symmetrical - each object experiences a
gravitational force because of the field set up by any other
object.
The Gravitational Field
The gravitational field is a vector quantity equal
to the gravitational force acting on a particle
divided by the mass of the particle:
g
Fg on m 
m
The gravitational field equation can be used
to determine the value of g at any location
by:
GM
g
r
2
Escape Velocity
Suppose you want to launch a rocket vertically upward
and give it just enough kinetic energy (energy of
motion) to escape the Earth’s gravitational pull.
The minimum initial velocity of an object at the Earth’s
surface that would allow the object to escape the Earth,
never to return, is the escape velocity.
Escape velocity from Earth:
v escape 
2  G  ME
rE
Satellite Orbits
A satellite is held in a circular orbit because the force of
gravity supplies the necessary centripetal force to keep
the object moving in a circular path about the central
body.
In order for a satellite to orbit around a central body,
such as the Earth, there must be a net force on the
object directed toward the center of the circular orbit, a
centripetal force.
For a satellite in orbit around Earth, the centripetal
force is equal to the gravitational force exerted by the
Earth on the satellite.
Satellite Orbits
A satellite does not fall because it is moving, being
given a tangential velocity by the rocket that launched
it. It does not travel off in a straight line because
Earth’s gravity pulls it toward the Earth.
The tangential speed of an object in a circular orbit is
given by:
v
G  ME
r
If the period of the orbit is known, the velocity may be
determined using:
2 r
v
T
Satellite Orbits
The period of a satellite can be determined
by:
2 r
T
v
Satellite Orbits
The Goldilocks principle can be used to explain the relationship
between the speed of a satellite and its orbit. The velocity of the
satellite is critical, and the velocity described by the equation:
v

r

g
min
describes the minimum velocity necessary for the satellite
Vmin
to maintain its proper circular orbit (JUST RIGHT). If the
satellite velocity is TOO HOT (greater than the vmin), it will not
maintain the proper circular orbit and fly into space. If the
satellite velocity is TOO COLD (less than vmin), it will be pulled
into the Earth’s atmosphere by the Earth’s gravitational force,
where it will either burn up in the atmosphere or slam into the
Earth’s surface.
Gravitational Potential
Energy Revisited
Gravitational potential energy near the
surface of the Earth is given by the equation
Ug = m·g·h, where h is the height of the
object above or below a reference level.
This equation is only valid for an object near
the Earth’s surface.
For objects high above the Earth’s surface,
the equation for potential energy is:
G  ME  m
U
r
Gravitational Potential Energy
The negative sign comes from the work done
against the gravity force in bringing a mass in
from infinity where the potential energy is
assigned the value zero, towards the Earth.
This work is stored in the mass as potential
energy.
As r gets larger, the potential energy gets
smaller; the gravitational force approaches
zero as r approaches infinity.
Gravitational Potential Energy
Only changes in gravitational potential
energy are important.

For an object that moves from point B to
point A, the expression for the change in
potential energy is:
 G  ME  m   G  M E  m 

U  U A  UB 
 
rA
rB


Web Sites
Kepler's Laws (with animations)
Kepler's Three Laws
Kepler's Three Laws of Planetary Motion
Kepler's Laws of Planetary Motion