Download Chapter 14 THE LAW OF GRAVITY

Chapter 14 THE LAW OF GRAVITY
INTRODUCTION
In this chapter we study the law of gravity. Emphasis is placed on describing the motion of
the planets, because astronomical data provide an important test of the validity of the law of gravity.
We show that the laws of planetary motion developed by Johannes Kepler follow from the law of
gravity and the concept of the conservation of angular momentum. A general expression for the
gravitational potential energy is derived, and the energetics of planetary and satellite motion are
treated. The law of gravity is also used to determine the force between a partide and an extended
body.
EQUATIONS AND CONCEPTS
The universal law of gravity states that any
two particles attract each other with a force
that is proportional to the product of their
masses and inversely proportional to the
square of their separation.
The constant G is called the universal
gravitational constant.
The acceleration due to gravity, g*, decreases
with increasing altitude, h, measured from the
Earth*s surface.
A gravitational field g exists at some point
in space if a particle of mass m experiences a
gravitational force Fg = mg at that point. That
is, the gravitational field represents the ratio
of the gravitational force experienced by the
mass divided by that mass.
The gravitational field at a distance r from the
center of the Earth points radially inward toward
the center of the Earth. Over a small region near
the Earth*s surface, g is an approximately uniform
downward field.
G = 6.673 x 10-11 N m2/kg2
Since the gravitational force is conservative,
we can define a gravitational energy function
corresponding to that force. As a mass m moves
from one position to another in the presence of
the Earth*s gravity, its potential energy changes by
an amount given by Equation to the right, where
r1 and are the initial and final distances of the mass
from the center of the Earth.
The gravitational potential energy associated with
any pair of particles of masses m1 and m2 separated
by a distance r is given by Equation. The negative
sign in this expression corresponds to the attractive
nature of the gravitational force. An external agent
must do positive work to increase the separation of
the particles.
In this expression it is assumed that Ui = 0 at ri = 4,
and the equation is valid for an Earth-particle system
when r > RE.
As a body of mass m moves in an orbit around a
very massive body of mass M (where M>> m),
the total energy of the system is the sum of the
kinetic energy of m (taking the massive body to
be at rest) and the potential energy of the system.
When the two contributions are evaluated for a
circular orbit, one finds that the total energy E is
negative, and given by equation. This arises from
the fact that the (positive) kinetic energy is equal
to one half of the magnitude of the (negative)
potential energy.
The escape velocity is defined as the minimum
velocity a body must have, when projected from
the Earth whose mass is ME and radius is RE, in order
to escape the Earth*s gravitational field (that is, to just
reach r = 4 with zero speed). Note that Vesc does not
depend on the mass of the projected body.
The potential energy associated with a particle of
mass m and an extended body of mass M can be
evaluated using equation, where the extended
body is divided into segments of mass dM, and r is
the distance from dM to the particle.
If a particle of mass m is outside a uniform solid
sphere or spherical shell of radius R, the sphere
attracts the particle as though the mass of the sphere
were concentrated at its center.
If a particle is located inside a uniform spherical
shell, the force acting on the partide is zero.
If a particle of mass m is inside a homogeneous solid
sphere of mass M and radius R, the gravitational
force acting on the particle acts toward the center
of the sphere and is proportional to the distance r
from the center to the partide. This force is due only
to that portion of the mass contained within the
portion of the sphere of radius r < R.
SUGGESTIONS, SKILLS, AND STRATEGIES
In this chapter we made use of the definite integral in evaluating the potential energy function
associated with the conservative gravitational force. You should be familiar with the following type
of definite integral:
For example we used the above expression as follows:
REVIEW CHECKLIST
•
State Kepler*s three laws of planetary motion and recognize that the laws are empirical in
nature; that is, they are based on astronomical data. Describe the nature of Newton*s
universal law of gravity, and the method of deriving Kepler*s third law (T2 % r3) from this
law for circular orbits. Recognize that Kepler*s second law is a consequence of conservation
of angular momentum and the central nature of the gravitational force.
•
Understand the concepts of the gravitational field and the gravitational potential energy, and
know how to derive the expression for the potential energy for a pair of partides separated
by a distance r.
•
Describe the total energy of a planet or Earth satellite moving in a circular orbit about a
large body located at the center of motion. Note that the total energy is negative, as it must
be for any dosed orbit.
•
Understand the meaning of escape velocity, and know how to obtain the expression for vesc
using the principle of conservation of energy.
•
Learn the method for calculating the gravitational force between a particle and an extended
object. In particular, you should be familiar with the force between a particle and a spherical
body when the partide is located outside and inside the spherical body.