**Gravity**

Welcome back! There are only a few more topics

that we need to complete before the end of the

semester.

We are just going to make some comments on

Newton's universal law of gravitational attraction.

A bit about potential energy.

And a bit from Kepler too.

The first topic is a closer look at the force of

gravity and the acceleration due to gravity.

We have been using the force of gravity/acceleration

due to gravity as F=mg most all the semester.

And have met with success.

This is because the value at the surface of the earth

doesn't fluctuate very much.

If we compare this form with Newton's Universal

law of gravity we can note the relationship:

Solving for everything but the little mass:

The clever thing about this equation is that it

would allow us to find the acceleration due

to gravity on the surface of any planet

The obvious extension to this is finding this

free-fall acceleration at other distances as well

Taking a different look at Newton's universal equation

for the force of gravity, we can solve for the mass of the

earth.

As you recall we discussed Cavendish's experiment, how he used a torsion wire to measure the value of G.

With that value we can solve for the mass of the earth, by

rearranging the previous equation:

This comes because of the constant G, it is one of a few universal constants.

**Potential energy**

You may recall from previous chapters the relationship

between force and energy

I drew a graph on the board about potential energy and

took the negative slope as the force.

Then I believe we looked at taking the integral of the

force and this lead to the potential energy from gravity

Note it was the negative slope, and hence the negative

integral. Doing this with the universal force of gravity:

Escape velocity

This is a bit of a tricky problem conceptually

If we consider only a rocket and the planet

then when the rocket has left the atmosphere

it will still be feeling the effects of gravity, and will

fall back to the earth.

So.... we let it go to an infinite distance, and when

it finally gets there we'll say it can finally stop

using this we look at the initial and finally energies

Solve for velocity

**Satellites and orbits**

We've looked at the motion of objects moving in a

circle before, and said there must be a force

pulling them to the center of that circular path.

As we consider this force the universal force of gravity

we can solve for an orbital velocity as a function of the

distance

Again solving for velocity

Kepler's third law related period to radial distance

We can solve for this law by using Newton's force

and the definition of velocity from period.

Velocity is distance over time. If this distance is one

revolution then the time is the period.

This then is Kepler's third law.

You'll note it relates the period of rotation around

a mass M at a distance of r.

We can use it to relate any orbital period around

the same mass, as in all the planets around the sun.

In addition we can look at geosynchronous orbits.

This is done by setting the period of revolution to

the period of rotation.

For the earth this distance is

The last concept to consider is the energy in an orbit

we said that the velocity needed to move in a circular

path could be found from

Solving for velocity

Now take this velocity and put it into a kinetic equation

As compared to potential energy

And finally total mechanical energy

Note this is a negative total energy, and this is what is

called a bound system, the potential energy is negative

not just zero, a satellite can't escape

**We could do any number of examples of**

any of these problems.

In truth we are just using one or two equation, and then sticking that equation into some other

previously known equations.

any of these problems.

In truth we are just using one or two equation, and then sticking that equation into some other

previously known equations.

**Bit of review**

"It wasn't really clear to me in the reading what gravitational potential energy was between to objects. I get it from the previous lessons when you had something just above earth's surface but how does it apply to just two objects and how they affect one another?"

"Can we go over the equation we use to solve gravitational potential energy problems? What does it mean and how do we use it?"

"I couldn't understand very well how the earth was weighed."

"can you explain more about the differences between little g and big G?"

A starship is circling a distant planet of radius R. The astronauts finds that the free-fall acceleration at their altitude is half the value at the planets surface. How far above the surface are they orbiting?

.41 R

A projectile is shot straight up from the earth's surface at a speed of 10,000 km/h. How high does it go?

394 km

418 km

(compare to flat earth model)