1425 Gravitation
Physics Lecture 11
Michael Fowler, UVa
The Inverse Square Law
Newtons idea: the centripetal force keeping the
Moon circling the Earth is the same gravitational
force that pulls us to the ground.
BUT: the Moons centripetal acceleration is only
1/3600 of g!
The Moon is 60 times further from the Earths
center than we are.
SO: this suggests the gravitational pull from the
Earth decreases with distance from the Earths
center as 1/r2.
Gravitational Attraction is Proportional to Mass
The gravitational force causes everything to
fall with the same acceleration:
For downward motion Fgrav = ma = mg
That is, Fgrav/m has the same value for all
masses.
Conclusion: F m: the gravitational force on
a mass is proportional to the mass.
Clicker Question
Does the Moon attract the Earth gravitationally?
A. No
B. Yes, but with a much weaker force than the
Earth attracts the Moon.
C. Yes, with the same force the Earth attracts
the Moon.
So why doesnt the Earth go round the
Moon?
The moons mass is about 1% of the Earths
mass.
In fact, the Moons attraction does cause the
Earth to go in a circleboth Earth and Moon
circle their common center of mass (which is
inside the Earth!) Well discuss this more
later.
Gravitational Attraction is Proportional
to Both Masses
Weve already seen, from Galileos
observation of equal downward acceleration
and Newtons Second Law, that the Earth
attracts an object with a force proportional to
the objects mass.
But from Newtons Third Law, the attractions
are equal and oppositesymmetric.
Therefore, the attraction is also proportional
to the Earths mass.
Law of Universal Gravitation
The story so far:
For two masses m1, m2 at a distance r apart, the
gravitational attraction between them is
proportional to 1/r2
It is also proportional to both m1 and m2.
Therefore it must have the form
m1m2
F =G 2
r
where G is some constant.
Clicker Question:
How could G be determined?
A. By accurately measuring the fall of a carefully
weighed object
B. By observing the satellites of Jupiter
C. From calculations based on measuring relative
planetary motions in the Solar System
D. It cant be determined in any of these ways
without some further information
Measuring G
G was first measured
in 1798 by a wealthy
English aristocrat,
Cavendish. It was a
very expensive
experimentthe
apparatus is ten feet
high, in its own
constructed room. He
measured a
10-7 Newton attraction
between lead balls.
http://www.youtube.com/watch?v=vWlCm0X0QC0
http://www.youtube.com/watch?v=EE9TMwXnx-s&NR=1
Weighing the Earth
Cavendish called his experiment
weighing the Earth: he knew the
inverse square law, the big balls had
masses of about 150 kg, and were
about 0.25 meters away from the small
balls.
Comparing the attraction of the small
balls to the big balls with the small
balls attraction to the Earth (the ratio
was about 10-8) and allowing for the
different rs (ratio about 3x106) he
found the mass of the Earth to be
6x1024 kg, with about a 2% error from
the known modern value.
The value of G is 6.67x10-11.
Cavendish s measurements
give G = 6.75x10-11.
Weighing the Sun
Once we know G, we can find the mass of the Sun.
Taking the Earths orbit around the Sun to be a circle,
the Suns gravity providing the centripetal force, F =
ma is
mv 2 GMm
=2
r
r
simplifying to
GM = v 2 r.
Putting in v = 30 km/sec , r = 150x106 km, G =6.7x10-11,
we find M = 2x1030 kg.
Weighing a Galaxy
We can estimate the
mass of a galaxy by
measuring the
centripetal acceleration
of an outer star. We can
also estimate it by just
counting starsbut the
mass turns out to be
much greater that the
total mass of visible
stars. Most of the mass
is dark matter.
http://geology.com/nasa/nasa-universe-pictures.shtml
Vector Form of Gravitational Force
The gravitational force is
of course a vector, the
attraction of sphere B on
sphere A points from the
center of A towards that
of B,
GmAmB
FAB =
rAB
2
rAB
where rAB is a unit vector
A
A
B
C
pointing from A to B.
The total force on A is = FAB + FAC .
FA
Massive Spherical Shell
Imagine a massive hollow
uniform spherical shell.
The gravitational is force the sum
of the attractive forces from all
parts of the shell.
Inside the shell, this force is zero
everywhere! Smaller closer areas
balance larger more distant
areas.
Outside the shell, Newton proved
the force was the same as if all
the mass were at the center.
A
Clicker Question
How will g change (if at all) on going from
the Earths surface to the bottom of a deep
mine?
A. g will be a bit stronger at the bottom of the
mine
B. It will be weaker
C. It will be the same as at the surface
Keplers First Law
Each planet moves in an elliptical orbit with the
Sun at one focus
He deduced this from
analyzing many
observations.
An ellipse is the set of
points P such that PF1 +PF2
is constant, the points F1
and F2 are the foci, PF1
means the distance from
point P to point F1.
A
F1
P
F2
You can draw an ellipse by fixing the ends
of a piece of string at A, B then, keeping the
string tight, loop it around a pencil point at
P and move the pencil around on paper.
About Ellipses
The standard notation is to
label the two foci F1, F2. (The
term focus is used because
if a light is placed at F1, and
the ellipse is a mirror, the
reflected light all goes to F2.)
The eccentricity e of the
ellipse is how far a focus is
from the center C compared
with the furthest point of the
ellipse.
e = 0 means a circle: most
planetary orbits are close to
circlesfor Earth, e = 0.017.
2b
A
F1
ea
C
P
2a
F2
2a is the length of the major axis,
2b the length of the minor axis
a is called the semimajor axis length.
Keplers Second Law
As the planet moves, a line from the planet to the center
of the Sun sweeps out equal areas in equal times.
Clicker Question:
The Earth is moving
fastest in its orbit when:
A. It is closest to the Sun
B. When its furthest from
the Sun
C. Cannot be determined
from this Law.
A
Sun
Animation!
P
Clicker Question
When is Earth closest to the Sun?
A.
B.
C.
D.
January
June
July
August
Keplers Third Law
Calling the average distance of a planet from the Sun
r, and the time for one complete orbit T, then
T2/r3 is the same number for all the Suns planets.
We can prove this for circular orbits (a good
approximation): F = ma is GMm / r 2 = mv 2 / r , and
putting in v = 2r/T gives
T
4
=
3
r
GM Sun
2
2
the same value for all the Suns planets!
Trip to Mars
The only practical
route to Mars is an
ellipse taking initial
full advantage of
the Earths own
orbital speed, and
just getting out far
enough to touch
Mars orbit. Timing
is crucial.
Try it here.
a
The Slingshot
The only way to really get out there is to use
the planets themselves to bump you up to
higher speed well, pull you up.
To see how Jupiter is used to get spacecraft
further out, check the slingshot animation.
Clicker Question: dropping an iPod!
An iPod has a built-in accelerometer, which gives a
continuous reading of gravity. Just resting on a table,
it reads g.
If I drop it, what does it read while falling?
A.
B.
C.
D.
Zero.
g.
Less than g, but not zero.
More than g.
Weightlessness in Orbit
We dont directly feel gravity, we feel the
compression of parts of our bodies, perhaps
stretching of other parts, supporting us.
The gravitational force on a body is just as
strong in a freely falling elevatorbut we
dont feel it.
An orbiting spacecraft is in freefall, just like
the elevator, so things in it are weightless in
the same way.
Footnote: Galileo already knew this
He was asked: if a large stone is falling with a small stone
on top of it, does the small stone press on the large one?
His reply: One always feels the pressure upon his shoulders
when he prevents the motion of a load resting upon him; but
if one descends just as rapidly as the load would fall how can
it press upon him? Do you not see that this would be the
same as trying to strike a man with a lance when he is running
away from you with a speed which is equal to, or even
greater, than that with which you are following him?
You must therefore conclude that, during free and natural fall,
the small stone does not press upon the larger and
consequently does not increase its weight as it does when at
rest.