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PH 121 13.4-13.6
Transcript of PH 121 13.4-13.6
Welcome back! There are only a few more topics
that we need to complete before the end of the
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
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.
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:
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
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.
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?
A projectile is shot straight up from the earth's surface at a speed of 10,000 km/h. How high does it go?
(compare to flat earth model)