GravitationCh 9: HasbunCh 5: Thornton & Marion

Introduction

Newton, 1666Published in Principia, 1687 (needed to develop calculus to prove his assumptions)

Newton’s law of universal gravitationEach 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 PotentialGravitational field vector can be written as the gradient of a scalar function:

Φ is the gravitational potentialEnergy/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 equilibriumL4 , L5

WMAP satellite in L2

MATLAB ProblemStart 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