Gravitation
Ch 5: Thornton & Marion
Introduction
Newton, 1666
Published in Principia, 1687 (needed to develop
calculus to prove his assumptions)
Newton’s law of universal gravitation
Each mass particle attracts every other particle in the
universe with a force that varies directly as the product
of the two masses and inversely as the square of the
distance between them.
Cavendish Experiment
Henry Cavendish (1731-1810) verified law and
measured G
G=6.67 x 10-11 N m2 / kg2
video
Extended Objects
Gravitational Field
Gravitational field = force per unit mass
For point masses:
For extended objects:
White Boards
Is gravity a conservative forces?
White Boards
Is gravity a conservative forces?
Gravitational Potential
Gravitational field vector can be written as the gradient
of a scalar function:
Φ is the gravitational potential
Energy/mass
We can obtain Φ by integrating:
Potential from Continuous
Mass Distributions
Prime denotes
integration element
Gravitational Potential
Once we know Φ, we can determine the gravitational
force and the gravitational potential energy.
Example
What is the gravitational potential both inside and
outside a spherical shell of inner radius b and outer
radius a?
Example
Astronomical measurements indicate that the orbital
speed of masses in many spiral galaxies rotating about
their centers is approximately constant as a function of
distance from the center of the galaxy. Show that this
experimental result is inconsistent with the galaxy
having its mass concentrated near the center of the
galaxy and can be explained if the mass of the galaxy
increases with distance R.
Poisson’s Equation
Gauss’s Law for the electric field
Gauss’s Law for gravity
Poisson’s Equation
Lines of Force & Equipotential Surfaces
Equipotential lines connect
points of constant potential
Force is always perpendicular
to the equipotential lines
Like a contour map, lines of
equipotential show where an
object can move while
maintaining constant
gravitational potential energy
Using Potential
Potential is a convenient way to calculate the force
Force is physically meaningful
In some cases, it might be easier to calculate the force
directly
Potential is a scalar
Example
Consider a thin uniform disk of mass M and radius a.
Find the force on a mass m located along the axis of
the disk. Solve this using both force and potential.
Lagrange Points
Solved by Euler & Lagrange
Sun is M1
Earth-Moon is M2
Stable equilibrium
L4 , L 5
WMAP satellite in L2
Ocean Tides
The Moon and Sun exert tidal forces on the Earth.
This is because the strength of the gravitational force
varies with distance, so that the near side of the Earth
feels a larger force or acceleration than the far side.
We can differentiate the gravitational force equation to
see how its strength varies over a distance dR.
Tides
Continuing: Multiplying both sides by dR yields
If we want to figure out differential force across the size
of the Earth, set dR = REarth . Then let d be the
separation between M and m.
Tides
Spring Tides occur when tidal forces from Sun and
Moon are parallel.
Neap Tides occur when tidal forces from Sun and
Moon are perpendicular.
Moon returns to upper transit 53 minutes later each
day, so high tide occurs approximately 53 minutes later
each day.
White Boards
In the early 1980's the planets were all located on the same
side of the Sun, with a maximum angular separation of
roughly 90 degrees as seen from the Sun. This rough
alignment was sufficient to make possible the Voyager
spacecraft grand tour. Some people claimed that this
planetary alignment would produce destructive earthquakes,
triggered by the cumulative tidal effects of all the planets.
Very few scientists took this seriously! To understand why,
compute the max tidal effects on Earth produced by Jupiter
(the most massive planet) and Venus (the closest planet).
Compare these tidal effects to those caused by the Moon
each month.
Solution
Compute ratios of tidal forces from Jupiter and the
Moon, and Venus and the Moon.
Mass of moon
7.3E22 kg
Dist to moon
3.84 E5 km
Mass of Venus
4.84 E24 kg
Dist to Venus
4.15 E10 m
Mass of Jupiter
1900 E 24 kg
Dist to Jupiter
6.3 E 11 m
MATLAB Problem
Start with the following code. Adjust the mass ratios and
contour levels until you recreate the plot showing the Lagrange
points. Name your file equipotential.m
Elegant Universe
Gravity- From Newton to Einstein
Rotation Curves of Galaxies
An example
Determine the radial profile of the enclosed mass and
the total mass within 8’.