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 ( ) verified law and measured G G=6.67 x N m 2 / kg 2 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 M 1 Earth-Moon is M 2 Stable equilibrium L 4, L 5 WMAP satellite in L 2

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 = R Earth. 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 moon7.3E22 kg Dist to moon3.84 E5 km Mass of Venus4.84 E24 kg Dist to Venus4.15 E10 m Mass of Jupiter1900 E 24 kg Dist to Jupiter6.3 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’.