Lecture 5 Newton -Tides ASTR 340 Fall 2006 Dennis Papadopoulos

Motion at constant acceleration a in meters/sec 2 Start with zero velocity. Velocity after time t is v(t)=at. The average speed during this time was v av =(0+at)/2=at/2 The distance traveled s=v av t=at 2 /2 Suppose you accelerate from 0 to 50 m/sec in 10 secs The distance s will be given by S=(1/2)(5 m/sec 2 )10 2 = 250 m The general formula if you start with initial velocity v(0) is s=v(0)t+(1/2)at 2 ACCELERATING MOTION

N1 with v  0 comes directly from Aristotle’s concept (object at rest remains at rest) by applying Galilean Relativity: change to frame with initial v=0 ; F=0 so object remains at rest; change frames back and v= initial v N3 is exactly what’s needed to make sure that the total momentum is conserved. So… Newton’s laws are related to the symmetry of space and the way that different frames of reference relate to each other. Conservation Principles

Action=Reaction If friction and pull balance exactly cart moves with constant velocity otherwise it slows down or accelerates depending on what dominates

Force and acceleration Forces between two bodies are equal in magnitude, but the observed reaction --the acceleration -- depends on mass If a bowling ball and ping-pong ball are pushed apart by spring, the bowling ball will move very little, and the ping-pong ball will move a lot  Forces in a collision are equal in magnitude, too

Velocity, as used in Newton’s laws, includes both a speed and a direction. V and also F and a are vectors. Any change in direction, even if the speed is constant, requires a force In particular, motion at constant speed in a circle must involve a force at all times, since the direction is always changing Circular or Elliptical Motion

What happens when there is no force

NEWTON’S LAW OF UNIVERSAL GRAVITATION Newton’s law of Gravitation: A particle with mass m 1 will attract another particle with mass m 2 and distance r with a force F given by Notes: 1. “G” is called the Gravitational constant (G=6.67  N m 2 kg -2 ) 2. This is a universal attraction. Every particle in the universe attracts every other particle! Often dominates in astronomical settings.

Gravitational Mass vs. Weight 3. Defines “gravitational mass” 4. Using calculus, it can be shown that a spherical object with mass M (e.g. Sun, Earth) gravitates like a particle of mass M at the sphere’s center.

Measuring G Gravitational forces Total force zero Same as if all the mass was at O

First Unification in Physics 1/3600 g

Apple falls 5 m in one sec Moon falls about 1.4 mm in one sec away from straight line Moon Earth R EM /R E =60 First grand unification

Inverse square law

Orbital and Escape Velocity V orb =7.8 km/sec V esc = 11 km/sec

V esc =(2GM E /R E ) 1/2

KEPLER’S LAWS EXPLAINED Kepler’s laws of planetary motion Can be derived from Newton’s laws Just need to assume that planets are attracted to the Sun by gravity (Newton’s breakthrough). Full proof requires calculus (or very involved geometry)

Planets natural state is to move in a straight line at constant velocity But, gravitational attraction by Sun is always making it swerve off course Newton’s law (1/r 2 ) is exactly what’s needed to make this path be a perfect ellipse – hence Kepler’s 1 st law.(use calculus) The fact that force is always directed towards Sun gives Kepler’s 2 nd law (conservation of angular momentum) Newton’s law gives formula for period of orbit

TIDES Daily tide twice Why? 1/R 2 law Moon Earth Water pulled stronger than the earth Earth pulled stronger than the water

TIDES Twice monthly Spring Tides (unrelated to Spring) and Twice monthly Neap Tides Sun moon Earth moon Full moon – extra low tides Earth New moon Extra high tides

TIDES Twice monthly Neap Tides Earth First Quarter Earth Last quarter Sun moon at right angles