Transcript Newton`s Law of Gravity - d_smith.lhseducators.com

Newton’s Law of Gravity
I wonder
what’s for
lunch?
Where did it come from?
Newton was watching the full
moon one day, wondering why
inertia didn’t cause the moon to fly
off into space.
He realized that some force must
be holding the moon near to the
earth.
Where did it come from?
When an apple fell near Newton,
he realized that the force pulling
the apple down towards the earth
was the same force holding the
moon in its orbit.
Newton realized 3 things:
There was a force of attraction
between the earth & the moon, and
the force was somehow related to
the earth’s mass.
Hey, baby,
I’m attracted
to you!
Newton realized 3 things:
 From his 3rd law of motion, Newton
knew that if the earth was pulling on
the moon, the moon was also pulling
on the earth equally hard.
 Therefore, the force of gravity also
depended on the mass of the object
orbiting the earth (the moon.)
Newton realized 3 things:
Just like a magnetic force, as the
distance between the two masses
increased, the force of gravity
would grow weaker, not slowly &
evenly, but very quickly. Think
about how magnets attract each
other.
The Inverse Square Law
As the distance between the 2
objects increases, the force of
gravity decreases with the square
of the distance.
2
F is proportional to 1/d , not just
1/d.
Inverse Square Law
If you double
the distance
between 2
objects, the
force of gravity
between them
shrinks to ¼.
F=1
F=¼
Inverse Square Law
If you triple the distance between 2
objects, the force of gravity
2
between them shrinks to (1/3) or
th
1/9 .
It works the opposite way when 2
objects move closer together…
Inverse Square Law
If the distance between the earth &
the moon somehow were decreased
to ½ what it is now, the force of
gravity between the earth & the
moon would increase to 4 times
stronger than it is right now.
Let’s put it all together
F is the force of gravity.
F = G x Massearth x Massmoon
(distance between them)2
The ‘G’ in the formula is just a
factor that makes all the units work
out correctly.
Law of Gravity Equation
The equation can be applied to any
2 objects in space.
F = G x Mass1 x Mass2
(distance)2
G is a constant (a scaling factor)
equal to 6.67 x 10
-11
2
Nm /kg
2
Consequences for Astronomy
This means that every object in the
universe, every planet, star, even
hydrogen atom, attracts every other
object in the universe.
When the 2 objects are very far
apart, the attractive force is very
small, but it’s still present.
I can still feel
you, even way
over there!
Gravity never sleeps!
Consequences for Astronomy
The gravity law explains why
planets orbit stars…
why stars orbit the center of the
galaxy…
why all the galaxies in the universe
should be attracted to one another.
Some examples
The acceleration due to gravity is
9.8 m/s at the earth’s surface.
If you climbed to the top of the
highest mountain, you would be a
little further from the earth’s
center. An object dropped here
would weigh a little less and drop
slightly slower than it would at the
earth’s surface.
2
What would happen if…?
If the earth suddenly shrank to ½
its current size…the acceleration
due to gravity would be 4 times
what it is now.
You would feel 4 times heavier.
I simply MUST
lose some
weight!
What would happen if…?
If you visited a planet that was the
same size as the earth, but had
twice the mass…you’d feel twice
as heavy, and you’d accelerate
twice as fast in a fall.
What would happen if…?
If you visited Mars, where the
gravity is less than it is on
earth…you’d be able to lift 2.5
times what you can lift on
earth…you’d be able to throw a
baseball 2.5 times farther.
What would happen if…?
…if you were in a spaceship
orbiting a star, and the star
suddenly shrank to become a black
th
hole only 1/1000 of its former
size?
Would you be instantly ‘sucked
in?’
What would happen if…?
 Neither your mass, nor the mass of
the star has changed.
 Your distance from the center of mass
of the star hasn’t changed.
 The force of gravity between you and
the new black hole would be exactly
the same as it was before the star
became a black hole. You would
NOT be sucked in.
But, I thought…
 But don’t black holes have enormous
gravity?
 Yes, they do…at their surfaces!
 If you were standing on the surface
of the star when it shrank and
became a black hole, you would be
instantly crushed by the increase of
gravity.
I’m confused…
 Look at the law of gravity.
F = G x Mass1 x Mass2
2
(distance)
 The only thing that changes between
when the star is large and when it
shrinks to become a black hole is the
distance between you (at its surface)
and the center of the star’s mass.
Uh…OK…maybe
In the equation, the distance factor
between you and the black hole’s
center, d, gets very small, making
the force of gravity very large.
Let’s add just a little more
Newton realized that his new law
of gravity could be combined with
rd
Kepler’s 3 law – the one that
relates the size of an object’s orbit
to its orbital period.
(period)2 = (orbit radius)3 or
p2 = a3
The combined equation looks
like…
p2 =
.
4 2 a 3
G (Mass1 + Mass2)
It may look complicated, but it’s
soooo useful.
What can this equation do?
There are 4 terms in the equation
that are ‘unknowns’. These are p,
a, Mass1 and Mass2.
If you are able to measure any 3 of
the terms, you can calculate the 4th
term.
What can this equation do?
The equation can be used to…
…calculate the mass of the sun
…calculate the mass of any planet
that has a moon
…look for planets orbiting other
stars
…discover new planets!
What can this equation do?
This equation was used by a
British astronomer, John Couch
Adams, and a French astronomer,
Urbain Leverrier, to predict the
position of the planet Neptune in
1845…a whole year before it was
ever observed with a telescope!
Wow! They ‘found’ a planet?
The 2 astronomers noticed that the
planet Uranus would sometimes
speed up, then slow down in its
orbit. They believed that this
change in speed was due to the
gravitational tug of another, more
distant planet. The other planet
was Neptune.
Let’s use the equation!!
The equation has been used to
calculate the mass of the sun,
starting with the orbit of the earth.
We know that the earth takes
365.24 days to make 1 orbit.
Converting this to seconds equals
31,600,000 seconds.
Keep going…
We also know that the earth orbits
the sun at a distance of 1 A.U. or
150,000,000 kilometers
Convert this distance to meters,
equals 150,000,000,000 meters.
Now, let’s re-arrange the equation.
A re-arrangement
.
p2 =
4 2 a3
G (Masssun + Massearth)
becomes
(Masssun + Massearth) = 4 2 a3
G p2
Keep going…
Now, if you realize that the mass of
the earth is tiny, compared to the
enormous mass of the sun, you can
just ignore Mearth in the calculation,
without being very far off in your
answer.
So the equation becomes
…it becomes
Masssun = 4 2 a3
G p2
If you’ve been paying attention…
If you’ve been paying attention,
and writing everything down to
this point, you now have enough
information to solve Newton’s
equation for the mass of the sun.
Go ahead, give it a try. It’s
question #17 on your homework!
Thanks for watching…
…now it’s homework time!