Transcript Integrative Studies 410 Our Place in the Universe

Newton’s Laws
How far away is the Moon?
• The Greeks used a special configuration of
Earth, Moon and Sun (link) in a lunar eclipse
• Can measure EF in units of Moon’s diameter,
then use geometry and same angular size of
Earth and Moon to determine Earth-Moon
distance
• See here
for method
That means we can size it up!
• We can then take distance (384,000 km)
and angular size (1/2 degree) to get the
Moon’s size
• D = 0.5/360*2π*384,000km = 3,350 km
How far away is the Sun?
• This is much harder to measure!
• The Greeks came up with a lower limit,
showing that the Sun is much further away
than the Moon
• Consequence: it is much bigger than the
Moon
• We know from eclipses: if the Sun is X times
bigger, it must be X times farther away
Simple, ingenious idea – hard
measurement
Timeline
Isaac Newton – The Theorist
• Key question:
Why are things happening?
• Invented calculus and physics
while on vacation from college
• His three Laws of Motion,
together with the Law of
Universal Gravitation, explain all
of Kepler’s Laws (and more!)
Isaac Newton (1642–1727)
Isaac Newton (1642–1727)
Major Works:
• Principia (1687)
[Full title: Philosophiae naturalis
principia mathematica]
• Opticks [sic!](1704)
• Later in life he was Master of the Mint,
dabbled in alchemy, and spent a great
deal of effort trying to make his
enemies miserable
Newton’s first Law
• In the absence of a net external force, a
body either is at rest or moves with constant
velocity.
– Contrary to Aristotle, motion at constant
velocity (may be zero) is thus the natural state
of objects, not being at rest. Change of velocity
needs to be explained; why a body is moving
steadily does not.
Mass & Weight
• Mass is the property of an object
• Weight is a force, e.g. the force an object of
certain mass may exert on a scale
Newton’s second Law
• The net external force on a body is equal to the
mass of that body times its acceleration
F = ma.
• Or: the mass of that body times its acceleration is
equal to the net force exerted on it
ma = F
• Or:
a=F/m
• Or:
m=F/a
Newton II: calculate Force from
motion
• The typical situation is the one where a
pattern of Nature, say the motion of a
planet is observed:
– x(t), or v(t), or a(t) of object are known, likely
only x(t)
• From this we deduce the force that has to
act on the object to reproduce the motion
observed
Calculate Force from motion: example
• We observe a ball of mass m=1/4kg falls to the
ground, and the position changes proportional to
time squared.
• Careful measurement yields:
xball(t)=[4.9m/s2] t2
• We can calculate v=dx/dt=2[4.9m/s2]t
a=dv/dt=2[4.9m/s2]=9.8m/s2
• Hence the force exerted on the ball must be
• F = 9.8/4 kg m/s2 = 2.45 N
– Note that the force does not change, since the
acceleration does not change: a constant force acts on
the ball and accelerates it steadily.
Newton II: calculate motion from
force
• If we know which force is acting on an object of
known mass we can calculate (predict) its motion
• Qualitatively:
– objects subject to a constant force will speed up (slow
down) in that direction
– Objects subject to a force perpendicular to their motion
(velocity!) will not speed up, but change the direction
of their motion [circular motion]
• Quantitatively: do the algebra
Newton’s 3rd law
•
For every action, there is an equal and
opposite reaction
•
Does not sound like much, but that’s
where all forces come from!
Newton’s Laws of Motion (Axioms)
1. Every body continues in a state of rest or in a
state of uniform motion in a straight line unless it
is compelled to change that state by forces acting
on it (law of inertia)
2. The change of motion is proportional to the
motive force impressed (i.e. if the mass is
constant, F = ma)
3. For every action, there is an equal and opposite
reaction (That’s where forces come from!)
Newton’s
Laws
Always the same constant pull
a) No force: particle at rest
b) Force: particle starts moving
c) Two forces: particle changes
movement
Gravity pulls baseball back to earth
by continuously changing its velocity
(and thereby its position)
Law of Universal Gravitation
Mman
MEarth
R
Force = G Mearth Mman / R2
Orbital Motion
Cannon “Thought Experiment”
• http://www.phys.virginia.edu/classes/109N/more_stuff/Appl
ets/newt/newtmtn.html
From Newton to Einstein
• If we use Newton II and the law of universal
gravity, we can calculate how a celestial object
moves, i.e. figure out its acceleration, which leads
to its velocity, which leads to its position as a
function of time:
ma= F = GMm/r2
so its acceleration a= GM/r2 is independent of its mass!
• This prompted Einstein to formulate his
gravitational theory as pure geometry.
Applications
• From the distance r between two bodies and the
gravitational acceleration a of one of the bodies,
we can compute the mass M of the other
F = ma = G Mm/r2 (m cancels out)
– From the weight of objects (i.e., the force of gravity)
near the surface of the Earth, and known radius of Earth
RE = 6.4103 km, we find ME = 61024 kg
– Your weight on another planet is F = m  GM/r2
• E.g., on the Moon your weight would be 1/6 of what it is on
Earth
Applications (cont’d)
• The mass of the Sun can be deduced from the
orbital velocity of the planets: MS = rOrbitvOrbit2/G
= 21030 kg
– actually, Sun and planets orbit their common center of
mass
• Orbital mechanics. A body in an elliptical orbit
cannot escape the mass it's orbiting unless
something increases its velocity to a certain value
called the escape velocity
– Escape velocity from Earth's surface is about 25,000
mph (7 mi/sec)