Transcript Friday, October 10

Algebraic Statements And
Scaling
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)
Scaling
• Often one is interested in how quantities
change when an object or a system is
enlarged or shortened
• Different quantities will change by different
factors!
• Typical example: how does the
circumference, surface, volume of a sphere
change when its radius changes?
How does it scale?
• Properties of objects scale like the
perimeter, the area or the volume
– Mass scales like the volume (“more of the same
stuff”)
– A roof will collect rain water proportional to its
surface area
Example from Homework:
Newton’s Law of Gravity
Note that in order to compute a "factor of change" you can ask: by what
factor do I have to multiply the original quantity in order to get the desired
quantity? Example: Q: By what factor does the circumference of a circle
change, if its diameter is halved? A: It changes by a factor 1/2 = 0.5, i.e.
(new circumference) = 0.5 * (original circumference), regardless of the value
of the original circumference.
• If the mass of the Sun was bigger by a factor 2.7, by what factor would the
force of gravity change?  scales linear with mass  same factor
• If the mass of the Earth was bigger by a factor 2.2, by what factor would
the force of gravity change?  scales linear with mass  same factor
• If the distance between the Earth and the Sun was bigger by a factor 1.2,
by what factor would the force of gravity change?  falls off like the
area  factor 1/ f 2 = 1/1.44 = 0.694