1. POWER POINT # 2
NEWTON’s LAWS
GRAVITY

2. I: NEWTON’S LAWS OF MOTION
Begin by stating Newton’s laws:
Newton’s first law (N1) – If a body is not acted upon by any forces, then its velocity, v, remains constant
Note:
N1 sweeps away the idea of “being at rest” as a natural state.

3. Newton’s 2nd law (N2) – If a body of mass M is acted upon by a force F, then its acceleration a is given by F=Ma
Note
N2 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 3rd law (N3) - If body A exerts force F on body B, the body B exerts a force –F on body A.

4. II: 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:
“G” is called the Gravitational constant (G=6.6710-11 N m2 kg-2)
This is a universal attraction. Every particle in the universe attracts every other particle! Often dominates in astronomical settings.

5. Defines “gravitational mass”
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.

6. 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)

7. 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

8. 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. N1) 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, N1 doesn’t hold – this is a non-inertial frame of reference.

9. ACCELERATING MOTION 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=at2/2
Suppose you accelerate from 0 to 50 m/sec in 10 secs
The distance s will be given by
S=(1/2)(5 m/sec2)102= 250 m
The general formula if you start with initial velocity v(0) is
s=v(0)t+(1/2)at2

10. Moon
Moon falls about 1.4 mm in one sec away from straight line
Apple falls 5 m in one sec
Earth
REM/RE=60

11. TIDES 1/R2 law Daily tide twice Why? Moon Earth Water pulled
stronger than
the earth
Earth pulled
the water

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

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

14. 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.

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

16. 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 vt
Centrifugal vs Centripetal vt a vr
The only real force is the
Centripetal force pulling
towards the center so the
ball must be accelerating o
An observer sitting on the ball feels a fictitious force Centrifugal that
pushes him outwards balanced by the string held at o.

17. 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= mvt2/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.

19. Maybe gravity is a fictitious force…
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?

20. Vesc=(2GME/RE)1/2

23. 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.