POWER POINT 2 NEWTONs LAWS GRAVITY I NEWTONS

POWER POINT # 2 NEWTON’s LAWS GRAVITY

I: NEWTON’S LAWS OF MOTION Begin by stating Newton’s laws: Newton’s first law (N 1) – If a body is not acted upon by any forces, then its velocity, v, remains constant Note: – N 1 sweeps away the idea of “being at rest” as a natural state.

Newton’s 2 nd law (N 2) – If a body of mass M is acted upon by a force F, then its acceleration a is given by F=Ma Note – N 2 defines “inertial mass” as the degree by which a body resists being accelerated by a force. – Another way of saying this is that force = rate of change of momentum (rate of change of mv). Newton’s 3 rd law (N 3) - If body A exerts force F on body B, the body B exerts a force –F on body A.

II: 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 10 -11 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.

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.

• I: 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

III: FRAMES OF REFERENCE We have already come across idea of frames of reference that move with constant velocity. In such frames, Newton’s law’s (esp. N 1) hold. These are called inertial frames of reference. Suppose you are in an accelerating car looking at a freely moving object (I. e. , one with no forces acting on it). You will see its velocity changing because you are accelerating! In accelerating frames of reference, N 1 doesn’t hold – this is a non-inertial frame of reference.

ACCELERATING MOTION 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 vav=(0+at)/2=at/2 The distance traveled s=vavt=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)102= 250 m The general formula if you start with initial velocity v(0) is s=v(0)t+(1/2)at 2

Moon falls about 1. 4 mm in one sec away from straight line Apple falls 5 m in one sec REM/RE=60 Earth

TIDES Daily tide twice Why? 1/R 2 law Earth Moon 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 Full moon – extra low tides Earth moon Sun Earth moon New moon Extra high tides

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

Real and fictitious forces In non-inertial frames you might be fooled into thinking that there were forces acting on free bodies. Such forces are call “fictitious forces”. Examples – • G-forces in an accelerating vehicle. • Centrifugal forces in fairground rides. • The Coriolis force on the Earth. Fictitious forces are always proportional to the inertial mass of the body.

Non – Inertial Frames • Monkey and hunter g • Accelerometer f a T Two real forces mg downwards and T. When car ma accelerates they must add to ma by Newton’s 2 nd law mg An observer in the car feels a force that pushes everything backwards. To explain his result

He adds a “fictitious force” known as inertial equal to Fin=-ma This is known as inertial force. T Sum of forces equal to zero ma mg Centrifugal vs Centripetal vt vt a vr The only real force is the o Centripetal force pulling towards the center so the ball must be accelerating An observer sitting on the ball feels a fictitious force Centrifugal that pushes him outwards balanced by the string held at o.

Look at the rotor in amusement park. People on the outside See only a Centripetal force from the wall pushing riders inwards into circular motion. People inside the rotor feel the fictitious Centrifugal force pushing them outward with the force from the wall balancing it. Centrifugal force Fc= mvt 2/R An observer in a non-inertial system with acceleration a should add to the real forces a fictitous inertial force Fin= -ma in order to describe the dynamics of the system correctly, i. e. to provide a description equivalent to an observer observing from an inertial frame. Weight-less-ness. Weight in an elevator or the shuttle.

Weak equivalence principle gravitational force is proportional to inertial mass. Maybe gravity is a fictitious force… … and we live in an accelerating frame of reference?

Vesc=(2 GME/RE)1/2

IV: NEWTON’S WORRIES • Newton knew that his theory has problems – Gravity is “action at a distance” – he didn’t like that! – A static universe would be gravitationally unstable.