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Gravitation
Ch 9: Hasbun
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?
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?
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
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