Gravitation
In galaxy M31, one very similar to our own, there is almost certainly a super massive black hole at the center. This very massive object supplies an enormous amount of gravitational pull.
Direction of Gravitation
Isaac Newton described this attractive force that acts between all pieces of matter in the universe.
Gravity is always attractive.
There is no “repulsive” gravity. So far, antigravity is just science fiction...
Law of Universal Gravitation
What factors do you think the gravitational attraction of two bodies would depend on?
mass of objects, and distance between objects
221
g d
mmF
Universal Gravitational Constant
The previous relationship only describes the factors that influence gravity. To get a numerically correct answer, with units, you need a constant added to the equation.
This constant is called the universal gravitational constant:
G = 6.67 x 10 -11 N m2 /kg2
Universal Law of Gravitation
221
g d
mmGF
Force from
gravity, N
Universal Gravitational
constant
Two masses,
kg
Distance between objects,
m
Because there is a quantity squared in the denominator of the fraction, this formula may be referred to as an inverse square law.
Gravitation Calculation Example
Calculate the gravitation force between the planet Mars, 6.4 x 10 23 kg, and the sun, 2 x 10 30 kg. Assume a distance of 2.0 x 10 11 m.
Fg
Gravitation Example Solution
221
g d
mmGF
Substitute values, do math carefully!
211
30232
211
g m)(2x10
kg)kg)(2x106.4x10)kgNm
(6.67x10F
Fg= 2.13 x 1021 N
Notice how the units cancel leaving the correct force unit of Newtons.
Gravitational Fields
Field lines show which way an object will move when exposed to some force.
Thus, gravitational field lines on earth always point to the center of the earth.
Many Gravitational Forces
Any object may have gravitational forces from different objects acting upon it.
The net gravitational force on any object is the sum of all the forces acting upon it.
n
2iiG FF
For however many objects you are considering, simply sum their influences to get the total gravitational force on an object.
Combining Forces
An astronaut is near two massive asteroids as shown. Describe the net gravitational force acting on him.
The exact magnitudes of the forces could be found if the masses and distances were known.
The net force is simply the sum of all gravitational influences acting on him.
F1
F1 Fnet
Thought Question
Imagine you are deep inside the earth in a cave. Would you weigh more, less, or the same as on the surface of the earth?
Thought Question Answer
When on the surface of the earth, all of the mass of the earth is pulling you down.
When underground, the mass above you actually pulls you up, countering some of pull from the bulk of the planet. You weigh less!
Extension of Thought Question
What would happen if you were at a cave at the center of the earth?
Because the gravitational pull from the matter equally all around you would cancel out, you would be weightless! Fnet would be 0.
Newton’s Shell TheoremThe previous two thought questions can be described with the shell theorem:
A uniform spherical shell of matter attracts an outside particle as if all the shell’s mass were
concentrated at its center.
Here we consider the earth to be many shells nested inside each other. The entire mass of the earth seems to be concentrated at its center.
More Shell Theorem
If you are inside the shell, that helps describe our cave questions:
A uniform spherical shell of matter exerts no net gravitational force on a particle located inside it.
No net influence from gravity inside the cave/shell.
Gravitational Field Inside a Planet
Notice how outside the planet the graph shows an inverse square law. Inside, it is linear. A maximum at the surface…
Planet Discovery
Gravity can have much more subtle effects.
The existence of Pluto and Neptune was predicted before they were ever observed.
Their slight gravitational effects wobble the orbits of the other planets, betraying their presence before they were ever seen visually.
Gravitational Potential Energy
In a previous class, we had a formula for the gravitational potential energy of an object here on earth:
mghPEgrav
Now that we have learned more about the universal nature of gravity, we will explore a more generic formula for gravitational potential energy.
This formula would apply to any two or more objects that gravitationally attract each other.
New Potential Energy Formula
d
mGmPE 21
grav
Universal Gravitational
constant
Two masses,
kg
Distance between
objects, m
You may notice some obvious similarities between this formula and the gravitational force formula.
Minus sign because the force and displacement are in
opposite directions
20
Ex: Imagine a mattress with a heavy bowling ball thrown in the middle. It forces the surface to conform around it.
Thus, Einstein would say that space is not a flat 2D grid like we often imagine.
Einstein’s Theory of Gravitation
The very existence of any massive body warps or bends space around it.
Gravity: Force or Curved Space
We typically think of gravity as a force. However, we could consider it a result of the curve that massive bodies create. Objects move as they roll along these curved created by massive objects.
A black hole is the ultimate in massive bodies curving space:
A black hole is often imagined like a funnel since it engulfs matter and crams it into an infinitely small spot called a singularity.
Questions?
Assignment: p 351 P # 1,18,24,29See inside text cover for some useful data