GravitationCh 5: Thornton & Marion

Integral for of Gauss’ Law

For spherically symmetric cases, g is in the radial direction, and so is n-hat

Example 1

Find the gravitational field for a spherical ring of mass with inner radius a and outer radius b. Find g for

r < a

a < r < b

r > b

Example 2

Find the gravitational field inside a uniform sphere of radius R.

Example 3

Find the gravitational field for a non-uniform sphere of radius R whose density is k/r2.

Find k

Find g for r < R

Find g for r > R

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 BoardsIn 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

Elegant Universe

Gravity- From Newton to Einstein

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.

Rotation Curves of Galaxies

An exampleDetermine the radial profile of the enclosed mass and the total mass within 8’.