# Gravitation - Siena College Gravitation Ch 9: Hasbun Ch 5: Thornton & Marion Introduction Newton, 1666 Published in Principia, 1687 (needed to

develop calculus to prove his assumptions) Newtons 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: is the gravitational potential Energy/mass We can obtain by integrating:

Potential from Continuous Mass Distributions Prime denotes integration element

Gravitational Potential Once we know , 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? Poissons Equation Gausss Law for the electric field Gausss Law for gravity

Poissons 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 , L5 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