Transcript newton`s laws of motion

POWER POINT # 2
NEWTON’s LAWS
GRAVITY
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.
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.
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
Gm1m2
Notes:
F
2
r
1. “G” is called the Gravitational constant
(G=6.6710-11 N m2 kg-2)
2. This is a universal attraction. Every particle in
the universe attracts every other particle! Often
dominates in astronomical settings.
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.
F
GMm
r2
•
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/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
2
4

2
3
P 
R
G ( M sun  M planet )
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.
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
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
TIDES
Daily tide twice Why?
1/R2 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
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 2nd 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= 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.
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=(2GME/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.