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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