Lecture 5: Gravity and Motion

Describing Motion and Forces

speed, velocity and acceleration momentum and force mass and weight Newton’s Laws of Motion conservation of momentum analogs for rotational motion

Torque and Angular Momentum

A torque is a twisting force Torque = force x

length of lever arm Angular momentum is

torque times velocity For circular motion,

L = m x v x r

Laws for Rotational Motion

Analogs of all of Newton’s Laws exist for rotational motion

For example, in the absence of a net torque, the total angular momentum of a system remains constant

There is also a Law of Conservation of Angular Momentum

Conservation of Angular Momentum during star formation

Newton’s Universal Law of Gravitation

Every mass attracts every other mass through a force called gravity

The force is proportional to the product of the two objects’ masses

The force is inversely proportional to the square of the distance between the objects’ centers

Universal Law of Gravitation

The Gravitational Constant G

The value of the constant G in Newton’s formula has been measured to be G = 6.67 x 10 –11 m3/(kg s2)

This constant is believed to have the same value everywhere in the Universe

Remember Kepler’s Laws?

Orbits of planets are ellipses, with the Sun at one focus

Planets sweep out equal areas in equal amounts of time

Period-distance relation:

(orbital period)2 = (average distance)3

Kepler’s Laws are just a special case of Newton’s Laws!

Newton explained Kepler’s Laws by solving the law of Universal Gravitation and the law of Motion

Ellipses are one possible solution, but there are others (parabolas and hyperbolas)

Conic Sections

Bound and Unbound Orbits

Unbound (comet)

Unbound (galaxy-galaxy)

Bound (planets, binary stars)

Understanding Kepler’s Laws:conservation of angular momentum

L = mv x r = constant

r

smaller distance smaller r bigger vplanet moves faster

larger distance

smaller v

planet moves slower

Understanding Kepler’s Third Law

42 a3 p2 =

G(M1 + M2)

Newton’s generalization of Kepler’s Third Law is given by:

42 a3 p2 =

GMsun

Mplanet << Msun, so

This has two amazing implications:

The orbital period of a planet depends only on its distance from the sun, and this is true whenever M1 << M2

An Astronaut and the Space Shuttle have the same orbit!

Second Amazing Implication:

If we know the period p and the average distance of the orbit a, we can calculate the mass of the sun!

Example:

Io is one of the large Galilean moons orbiting Jupiter. It orbits at a distance of 421,600 km from the center ofJupiter and has an orbital period of 1.77 days.

How can we use this information to find the mass of the Sun?

Tides

The Moon’s Tidal Forces on the Earth

Tidal Friction

Synchronous Rotation

Galactic Tidal Forces