Lecture 5 Newton -Tides

ASTR 340Fall 2006

Dennis Papadopoulos

Motion at constant acceleration a in meters/sec2

Start with zero velocity. Velocity after time t is v(t)=at.The average speed during this time was vav=(0+at)/2=at/2

The distance traveled s=vavt=at2/2Suppose you accelerate from 0 to 50 m/sec in 10 secsThe distance s will be given byS=(1/2)(5 m/sec2)102= 250 m

The general formula if you start with initial velocity v(0) is

s=v(0)t+(1/2)at2

ACCELERATING MOTION

• N1 with v0 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 m1 will attract another particle with mass m2 and distance r with a force F given by

Notes:1. “G” is called the Gravitational constant

(G=6.6710-11 N m2 kg-2)2. This is a universal attraction. Every

particle in the universe attracts every other particle! Often dominates in astronomical settings.

221

r

mGmF

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.

2r

GMmF

Measuring G Gravitational forces

Totalforcezero

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

REM/RE=60

First grand unification

Inverse square law

Orbital and Escape Velocity

Vorb=7.8 km/secVesc = 11 km/sec

Vesc=(2GME/RE)1/2

2( ) ( ) /

( ) ( )

g R GM r R R

Weight R mg R

2( ) ( ) /

( ) ( )

g R GM r R R

Weight R mg R

2( ) ( ) /

( ) ( )

g R GM r R R

Weight R mg R

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/r2) is exactly what’s needed to make this path be a perfect ellipse – hence Kepler’s 1st law.(use calculus)

The fact that force is always directed towards Sun gives Kepler’s 2nd law (conservation of angular momentum)

Newton’s law gives formula for period of orbit

32

2

)(

4R

MMGP

planetsun

TIDES

Daily tide twice Why? 1/R2 law

Moon

Earth

Water pulled stronger thanthe earth

Earth pulled stronger thanthe water

TIDES

Twice monthly Spring Tides (unrelated to Spring) andTwice monthly Neap Tides

Sun

moon

Earth

moon

Full moon – extra low tides

Earth New moon Extra high tides

TIDES

Twice monthly Neap Tides

Ear

th

First Quarter

Ear

thLast quarter

Sun moon at right angles