Key Concepts:

1. What is the Newton's Law of Gravity?
2. How are "mass" and "weight" different?
3. How is gravity used to weigh astronomical objects?
4. What causes the ocean tides?
5. What is tidal friction and how does it affect the Earth and the Moon?

Newton's Second Law of Motion

A net force is need to change the velocity of an object.

• Velocity = a measure of motion; a rate of change in position and direction (V = ΔX/Δt)


• Acceleration = a rate of change in velocity (a = ΔV/Δt)
• change in speed
• change in direction ("centripetal accelenration")

Law of Gravity

• Legendary experiment by Gallileo at the Leaning Tower of Pisa
• Q: What is the difference between "mass" and "weight"?


• Quiz 7A: Earth Diet


• Quiz 7B: Gravity on the Moon


• Quiz 7C: No Gravity in Space?

Masses from Orbital Speed

• Centripetal force: a force acting on any body moving in a circular path.

• For astronomical bodies in orbit around another object, centripetal force is balanced by gravitational force: F_C = F_g or

• By re-arranging the terms so that the mass of big object is on the left side and the rest are on the right side (see page 135 of the textbook):

Example: Mass of the Sun

• v = 30 km/sec = 3 x 10⁴ m/sec


• R = 1 AU = 1.5 x 10¹¹ m


• G = 6.7 x 10^-11


• Substituting these into M = v²R/G
• M = (3 x 10⁴)²(1.5 x 10¹¹) / (6.7 x 10^-11)
• M(Sun) = 2.0 x 10³⁰ kg

Newton's Version of Kepler's Third Law

• You can relate orbital speed and orbital period as


• By replacing v with this expression in the mass calculation above, you can show

"Weighing" Celestial Bodies using Orbital Motion

• Q1: What is the orbital period of a planet located 10 AU away from a Sun-like star?


• Q2: A planet is found around a nearby star X, located only 1 AU away. It takes the planet only 2 months to go around the star once. How much heavier is this star compared with the Sun?

Tides

• Quiz 7D: Tides of the Sun and Moon


• Ocean level rises falls twice a day.


• Earth stretched by the gravitational forces of the Moon.


• The Sun also exerts a tidal force on the Earth.
• Spring tides: The Sun and the Moon aligned


• Neap tides: The Sun and the Moon aligned perpendicularly




• Tidal Friction: Earth rotates against the stretch
• The Earth tidal bulge leads the Moon in rotation, and the Earth is slowed down by the Moon (a day is 1 second shorter in 50,000 yrs).


• In turn, the Moon is sped up and spirals outwards (conservation of angular momentum) - no more total solar eclipses in the distant future?


• Tidal friction circularizes orbits and synchronizes spins ("tidal locking") of satellites and close binary stars.

Quiz 7D: Tides of the Sun and Moon

Reading assignment for next lecture: Unit 20-22