Gravitation: Newtons Law

1
Gravitation Newtons Law
(Newtons Law of Gravitation)
(Classical Description of Gravity!)

• Gravity is an attractive force between two
  objects that have mass.

• The magnitude of the force of gravity is
  proportional to the product of the masses of the
  two particles (gravitational masses).
• The magnitude of the force of gravity is
  inversely proportional to the square of the
  distance between the two particles (i.e. 1/r2
  force).
• Newtons constant, G, is determined
  experimentally to be G 6.67 10-11 Nm2/kg2.

• Superposition.

The net gravitational force on mass m due to a
collection of masses, M1, M2, , MN is the vector
sum of the individual forces from each masses as
follows
2
Gravitational Potential Energy
Gravity is a conservative force which means that
we can define a gravitation potential energy DU
which is the amount of work done against the
gravitational force when the mass m moves from r1
to r2.
(gravitational potential energy)

• Superposition

Take r1 infinity and r2 r and define
U(infinity) 0 as the reference point!

• Example (escape speed)

A particle of mass m on the surface of a planet
with mass M and radius R is given an initial
velocity v0 at t 0. What is the minimum value
of v0 for the particle to escape the
gravitational attraction of the planet (i.e. not
get pulled back to the planet surface)?
(escape speed in any direction)
3
Gravitation Example

• Example

Two identical masses M are located on the y-axis
at y d and y -d. A third mass m is on the
x-axis a distance x from the origin.

• What is the net gravitational force, F(x), on
  the mass m due to the other two masses.

• Plot the magnitude of F(x) versus x.

• At what point xmax is F(x) maximum?

• What is the maximum value of F(x)?

• What is the gravitational potential energy U(x)?

• If the mass m is released from rest at x
  infinity, what is its speed when it reaches the
  origin?

4
Gravitation Overall Potential Energy

• Overall Potential Energy of a System Particles

Sum up the potential energy of each pair of
particles as follows
(overall potential energy, sum over all pairs)
Utot is the total amount of work (negative) done
against the gravitational force in bringing in
each mass from rest at infinity and placing it
(at rest) at the point rij. It is called the
binding energy. The minimum amount of (positive)
work necessary to separate the particles (from
rest) out to infinity is W -Utot.

• Overall Mechanical Energy of a System Particles

For isolated systems the total mechanical energy
is constant (i.e. Ei Ef).
(total energy)

• Example (binding energy)

Three identical masses M form an equilateral
triangle with sides of length L. What is the
binding energy of the system?
5
Gravitation Mass Distributions
If mass is distributed throughout some shape then
each infinitesimal piece of mass dM exerts a
force dFg on the mass m with

• Example (linear mass density)

A thin rod with uniform mass density and total
mass M and length L lies on the x-axis a distance
d from the origin. What is the magnitude of the
net gravitational force the rod exerts on a point
mass m at the origin?
6
Gravitation Uniform Spheres

• Outside a Sphere with Uniform Mass Density r

If mass is uniformly distributed throughout a
sphere of radius R and total mass M then
(for r R same a point mass M at the origin)

• Inside a Sphere with Uniform Mass Density r

If the mass m is at a point r within a sphere
with constant mass density and total mass M the
force on it depends only on the amount of mass
within a sphere of radius r, M(r) as follows
(for r R)

• Spherical Shell with Uniform Mass Density r

If mass is uniformly distributed throughout a
spherical shell of inner radius R1 and outer
radius R2 and total mass M then
(for r R2 same a point mass M at the origin)
(for r lt R1)
7
Gravitation Circular Orbits (M gtgt m)
For circular orbits the gravitational force is
perpendicular to the velocity and hence does no
work and therefore the kinetic energy (and speed)
of the mass m is constant. The force Fg is equal
to the mass times the centripetal acceleration as
follows
(angular momentum, constant)
Assume M gtgt m so that the CM is at M!
(speed, constant)
(angular velocity, constant)
(binding energy)
For isolated systems, E and L are constant for
all gravitational orbits. For circular orbits r,
U, KE, v, and w are also constant.
(E lt 0 closed orbit)
(period of rotation)

• Summary (circular orbit radius r, M gtgt m)

8
Gravitation Circular Orbits
Both masses rotate about the center-of-mass!

• General Solution (masses m and M)

The force Fg on the mass m is equal to its mass
times its centripetal acceleration as follows
(constant)
(angular velocity, constant)
(period of rotation)
(binding energy)

• Summary (circular orbit radius r)

(reduced mass)
9
Circular Orbits Examples

• Example 1

Consider two equal masses a distance R apart that
are undergoing circular motion due to their
gravitational attraction. In case 1 one of the
masses is held fixed at the origin and the other
revolves around it. In case 2 both masses are
free to move and they revolve around their
center-of-mass. Compare the energy E,
gravitational potential energy U, kinetic energy
KE, angular momentum L, angular velocity w, speed
v, and period T, for the two cases.

• Example 2

Two diametrically opposed masses m revolve around
a circle of radius R. A third mass M 2m is
located at the center of the circle. What is the
energy E, gravitational potential energy U,
kinetic energy KE, and period T of rotation for
this system of three masses.
10
Gravitation Elliptical Orbits

• Properties of the ellipse

An ellipse has two focal points F1 and F2
separated by a distance 2ea. For any point P on
an ellipse r r2 2a (constant), where r is the
distance between F1 and P and r2 is the distance
between F2 and P.

• e is the eccentricity.
• a is the semi-major axis.
• b is the semi-minor axis.
• Area A pab.
• rmin a(1-e) the perihelion point.
• rmax a(1e) the aphelion point.

(0 e lt 1)

• Elliptical Orbits (M gtgt m)

Planets and satellites with mass m move in
elliptical orbits with mass M at one focus.
(angular momentum, constant)
(Kinetic energy not constant)
(energy, constant)
11
Gravitation Elliptical Orbits

• Elliptical Orbits (M gtgt m)

Consider the area dA swept out by the line
between m and M in going from q to q dq.
(constant)
where w dq/dt is the angular velocity. The line
between m and M sweeps out area at a constant
rate (consequence of angular momentum
conservation).
(0 e lt 1)

• Period of the Orbit (M gtgt m)

Now consider the amount of area swept out in
going from q 0 at t 0 to q 2p at t T (i.e.
one period).
Area of ellipse!
12
Gravitation Elliptical Orbits

• Energy Conservation (M gtgt m)

The overall energy is constant and hence the
energy at perihelion, rmin, is equal to the
energy at the aphelion, rmax, and hence
(constant energy)
At perihelion and aphelion v is perpendicular to
r (i.e. vr 0) and hence
where
hence
and
The energy E and the period T depend only on the
semi-major axis a of the ellipse!
13
Gravitation Open and Closed Orbits

• Trajectories (M gtgt m)

v
Let v be the speed of a mass m when it is a
distance r R from the Earth (mass M) and let v
be perpendicular to r. Let v av0, with
(circular orbit radius R)
If v v0 (i.e. a 1) then the mass m will
undergo circular motion with energy E0 lt 0.
(circular orbit radius R)
In general we have
14
Equivalence of Inertial and Gravitational Mass

• Inertial Mass

The inertial mass of an object tells us how
difficult it is to change a particles velocity
(i.e. accelerate it). One can uniquely determine
the inertial mass m of a particle relative to the
reference mass mref by applying an equal force to
m and mref and measuring the ratio of the
resulting accelerations as follows
Inertial Mass

• Gravitational Mass

The gravitational mass mg of a particle is
determined (at the surface of the earth) by
measuring the gravitational force exerted on it
as follows
Gravitational Mass

• Experimental Observation (a1 a2)

All objects fall with the same acceleration!
We see that mi/mg is the same for all particles
and hence we can define G such that mi mg for
all particles. Only one type of mass!
15
Equivalence Principle

• Equivalence Principle (prelude to general
  relativity)

It is not possible to distinguish by any
experiment whatsoever between an accelerated
frame and an inertial frame with an appropriate
gravitational field.

• Frame 1 (at rest with gravity)

Consider a mass mg (at rest) on the end of a
spring with spring constant k in a gravitational
field. Thus,

• Frame 2 (accelerated, no gravity)

Now suppose the mass is on the end of the spring
with spring constant k in an accelerated frame
with acceleration a g and no gravity. In this
frame,
The equivalence principle tell us that the
observations in the two frames have to be
identical and hence d1 d2 which implies that mi
mg!
16
The Deflection of Light by Gravity
The equivalence principle predicts that light is
deflected by a gravitational field.

• Frame 1 (at rest with gravity)

Consider a light ray traveling through a hole on
the left and right side of a box of width W. The
holes are located a height h above the floor and
the box is at rest in a gravitational field with
gravitational acceleration g GM/R2.

• Frame 2 (accelerated, no gravity)

Suppose that the light entered the left hole at t
0. It takes the light a time t W/c to cross
the box. At time t the box has speed v gt and
has moved upward a distance d gt2/2. In frame
2 the light hits the right wall of the box a
distance d below the right hole!
The same thing must occur in both frames!
(Light Path)
Even though light is massless its path is bent by
gravity!