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