Transcript power point file

5. Universal Laws of Motion
“If I have seen farther than others, it
is because I have stood on the
shoulders of giants.”
Sir Isaac Newton
(1642 – 1727)
Physicist
Image courtesy of NASA/JPL
Sir Isaac Newton (1642-1727)
• Invented the reflecting
telescope
• Invented calculus
• Connected gravity and
planetary forces
Philosophiae naturalis
principia mathematica
The Acceleration of Gravity
• As objects fall, they
accelerate.
• The acceleration due to
Earth’s gravity is 10 m/s
each second, or g = 10
m/s2.
• The higher you drop the
ball, the greater its
velocity will be at
impact.
• Gravity of the moon is
1/6 as much..on Mars
its 1/3…
• Why?
The Acceleration of Gravity (g)
• Galileo demonstrated that g is the same for all objects,
regardless of their mass!
• This was confirmed by the Apollo astronauts on the
Moon, where there is no air resistance.
Is Mass the Same Thing as Weight?
• mass – the amount of matter in an object
• weight – a measurement of the force which
acts upon an object
What is your mass on
the Moon?
What is your weight on
Mars?
Do you weight the
same on a tall mountain
as at Sea level?
Weight and Mass
Image courtesy of NASA/JPL
In Free fall, you are weightless
Newton’s Laws of Motion
1 A body at rest or in motion at a constant
speed along a straight line remains in that
state of rest or motion unless acted upon
by an outside force.
If you are moving at constant speed in a circle, are you
accelerating?
Centripetal Force
• Don’t forget
centripetal
acceleration:
• In a circle,
• V = 2pr/T
• So F = 4p2rm/T2
2
v
ac 
r
so
2
v
Fc  m
r
Newton’s Laws of Motion
2 The change in a body’s velocity due to an
applied force is in the same direction as
the force and proportional to it, but is
inversely proportional to the body’s mass.
F/m = a
is the ratio of F/m always g? Why doesn’t
m matter?
New force law?
•
•
•
•
Connect the dots
1- F/m independent of m, so…
F = something x m
Acceleration of moon towards earth = .0027m/s2
so…
• So, that’s g/3600!
• So, that’s g/602
• So, the moon is 60 Earth radii away! So….
More connections
• So among = g/(60R)2
So a proportional to 1/distance2
Since a = F/m,
F proportional to 1/distance2
So what do we have?
F = something x m/d2
Finally Force is an interaction between two masses!
I.e. Ball pulls on Earth and Earth pulls on ball
So….F proportional to Mem/d2
Universal Law of Gravitation
Between every two objects there is an
attractive force, the magnitude of which is
directly proportional to the mass of each object
and inversely proportional to the square of the
distance between the centers of the objects.
Newton, Kepler, Galileo
• Newton used Galileo’s law of Inertia and…
• Galileo’s formula for calculating centripetal
acceleration and…
• His formula relating Force and Accleration
to
• Derive Kepler’s laws
• Formulate the law of Universal gravitation
Copernicus (1473 - 1534)
• by 1400 the planetary positions were no
longer predicted by the “almagest”
• Copernicus Proposed all the following
“fix”:
1. Earth spins on its axis once every 23
hrs, 56 min
2. Earth and all known planets orbited the
sun in circular orbits with sun at center.
3. distant stars were so far that no parallax
could be seen .
4. Polar axis precessed every 26,000
years.
5. All the above just a mathematical model
to make accurate predictions…easier
than updating the Ptolemaic model
So was Copernicus or Ptolemy’s
model
correct?
•Tycho Brahe, Johanes Kepler, and
Galileo were the greatest contributors
to the debate.
•Brahe (pronounced Bray) was the last
and greatest pre-telescopic astronomer.
•Brahe felt that better observations
were needed .
•interested in proving that the “Tyconic
Universe" was correct
Johannes
Kepler
(1571
1630)
•Mathematician...sought out Brahe
for his famous data, and was hired
by Brahe to fit data to the tyconic
model universe. But….
•Kepler was trying to fit data to the
“Kepler model" Universe
•the orbits of the six known planets
fit into the largest spheres which
could be inscribed into the six
regular geometric solids
•--crazy by today's standards, but at
least the orbits were centered on the
sun!
•Wrote: Harmony of the worlds
relating music, geometry, astronomy
Kepler and Brahe...continued
•Brahe died and family wouldn't
release data after Brahe died
(don’t ask how he died). :
•Eventually Kepler "acquired"
Brahe's data and found that:
•--the orbit of Mars just isn't a
circle!
•Plato was wrong!
•The door to a true
understanding of the solar
system was now wide open!
Brahe’s tombstone, from: www.nada.kth.se/~fred/tycho/tychotomb.jpg
•
Kepler’s
results
From Brahe’s data, Kepler
deduced three laws:
1. Planets orbit the sun in
Ellipses with the sun at one
focus
2. A line joining a planet and
the sun sweeps out equal
areas in equal times
These three relations are now
known as Kepler's three laws.
Extra for experts: x and y are the positions
of the Earth with 0,0 at intersection of
major and minor axis.
Kepler’s
third
law
1. The cube of a planets semimajor axis is proportional to
the square of its orbital
period a3 = T2
2. a = semi-major axis –also
average distance from
planet to sun (also written as
d or r). Units are AU’s.
3. T = orbital period (also
written as P). Units are
years.
4. Newton later used this
A graph of semi-major axis cubed on
discovery to develop and
vertical axis, and orbital period squared on
prove the law of universal horizontal axis. Clearly not a coincidence!
gravitation.
Newton’s Version of Kepler’s Third Law
Using the calculus, Newton was able to derive
Kepler’s Third Law from his own Law of Gravity.
In its most general form:
2
2
3
P = 4p a / G (m1 + m2)
If you can measure the orbital period of two
objects (P) and the distance between them (a),
then you can calculate the sum of the masses
of both objects (m1 + m2).
Determining the Mass
Image courtesy of NASA/JPL
4p 2 3
a
GM
where M is the larger mass
Often one mass is so much
smaller than the other, the small
mass can be ignored
so...P 2 
G is the constant of Universal Gravitatio n
N  m2
kg 2
Constants determie the units used
Newton didn' t know G...so he used ratios :
G  6.67 x 10-11
2
3
 P1   a1 
     note the constants drop out!
 P2   a2 
Let P2 and a 2 be the period
and semi major axis of the Earth
or P2  1 year ,
and a 2  1AU
so P1   a 2  as long as the object is orbiting the sun!
3
3
Orbital Paths
• Extending Kepler’s
Law #1, Newton
found that ellipses
were not the only
orbital paths.
• possible orbital paths
– ellipse (bound)
– parabola (unbound)
– hyperbola (unbound)
Changing Orbits
orbital energy = kinetic energy +
gravitational potential energy
conservation of energy implies:
orbits can’t change spontaneously
An object can’t crash into a planet
unless its orbit takes it there.
An orbit can only change if it
gains/loses energy from another
object, such as a gravitational
encounter:
If an object gains enough energy so that its new orbit is unbound,
we say that it has reached escape velocity.
Tides
• Since gravitational force decreases with (distance)2, the Moon’s pull on
Earth is strongest on the side facing the Moon, and weakest on the
opposite side.
• The Earth gets stretched along the Earth-Moon line.
• Greatest force pulls water away from Earth towards moon.
• Both Earth orbits center of mass of Earth Moon System
• Weaker force allows water to slide away from Earth on side opposite
moon
Tidal Friction
• This fight between Moon’s pull & Earth’s rotation
causes friction.
• Earth’s rotation slows down (1 sec every 50,000 yrs.)
• Conservation of angular momentum causes the Moon
to move farther away from Earth.
Synchronous Rotation
• …is when the rotation period of a moon,
planet, or star equals its orbital period about
another object.
• Tidal friction on the Moon (caused by Earth)
has slowed its rotation down to a period of
one month.
• The Moon now rotates synchronously.
– We always see the same side of the Moon.
• Tidal friction on the Moon has ceased since
its tidal bulges are always aligned with Earth.
Escape Velocity
• If you want to leave
the planet, you have to
do work!


W   F (r )  dr
dr
r2
r1
F(r)
R
So you’d better have kinetic energy!


W   F (r )  dr
r2
r1
Note the tricky negative
signs!
1
GMm
2
E  mvo 
2
R
Leaving a sphere
•
•
•
•
(
GMm GMm

)
r1
r2
W=
r1= r, r2= ∞
GMm GMm

)
So W =  (
r

If you “escape” r =
infinity
GMm
)
• So W =  (
r
• Recall W = -DEp
• Ep = 0 at infinity.
F(r)
R
Energy conservation
• Ep(r)=
• Etotal =
• E=
(
GMm
)
r
1 2
GMm
mv  (
)
2
r
1 2 GMm
mv 
2
r
Lets escape finally
•
•
•
•
•
•
•
•
1
GMm
2
mvo 
2
R
At surface, E =
At r = ∞,
E=0+0
This means you’ve lost all
1
GMm
2
mv

0
escape
your speed getting to
2
R
infinity!
2GM
So Einitial = Efinal = 0
vescape 
R
For Earth, v0 is 11km/s
For Black hole this is c!
actually, newtonian
GM
mechanics don’t quite
v
blackhole  c 
escape
apply there…
R
Black Holes
•
•
•
•
For a Black Hole, we can use this to find R
R = GM/c2 = Radius of event Horizon
This works out to about 3km/solar mass
Largest black holes have about 3 billion
solar masses
• Smallest (so far) about 3 solar masses.