- Good morning.
All objects on this planet are gravitationally bound to it.
Notice when I jump up, that I do not leave the planet.
Flipping Physics
- He is assuming everybody who is watching this is actually on this planet, which is not necessarily true.
- Okay, if I were able to jump up fast enough, and therefore have enough mechanical energy.
I would be able to leave the planet.
The minimum amount of energy or work necessary to completely remove an object from another object
is called the binding energy.
One can also determine the binding energy of atoms and atomic particles.
In this lesson, we're determining the binding energy of an object to a planet.
We are actually going to start, however, with this simple example
of a book being lifted at a constant velocity through a short vertical distance h.
Bo, please draw a freebody diagram and sum the forces on the book
while it is moving upward at a constant velocity.
- The force you are applying to the book is up and the force of gravity is down.
The net force in the y-direction equals force applied minus force of gravity
and it also equals mass times acceleration in the y-direction.
Because the book is moving at a constant velocity, the acceleration in the y-direction of the book is zero.
Therefore, force applied equals force of gravity which equals the mass of the object
times the acceleration due to gravity.
- Billy, please determine the work done by the force applied on the book
while the book goes through a vertical displacement h.
- Work equals force times displacement times the cosine of the angle between the two vectors.
So, the work done by the force applied equals force applied times displacement times cosine theta.
Force applied equals mass times acceleration due to gravity.
Displacement equals h.
The angle between the direction of the force applied which is up and the displacement
which is also up is zero and the cosine of zero is one.
So, the work done by the force applied on the book equals mass times acceleration due to gravity
times h; the vertical displacement of the book.
- Notice, mgh or mass times acceleration due to gravity times vertical height above the horizontal zero line
equals the gravitational potential energy of an object in a constant gravitational field.
If we set the horizontal zero line at the initial height of the book,
then height initial equals zero and height final equals h then,......
the work done by the force applied to the book
equals the change in Gravitational
Potential Energy of the book.
- The work done on the book by the force applied is the same as the book's change in
Gravitational Potential Energy?
- Yeah, that makes sense. The force applied puts gravitational potential energy into the system.
- Okay, sure.
- Good. Okay, back to the concept of binding energy.
Remember the binding energy of a planet is the minimum amount of energy or work
necessary to completely remove an object from the planet.
In this case, the gravitational field is not constant. So Billy, what equation do we need to use for
Gravitational Potential Energy?
- When the acceleration due to gravity is not constant,
we need to use the equation for Universal Gravitation Potential Energy.
Which is capital U sub g equals the Universal Gravitational Constant times mass 1
times mass 2 divided by r squared.
- You forgot the negative and r is not squared!
I think you just gave us Newton's Universal Law of Gravitation not Universal Gravitational Potential Energy.
- Yeah, I'm sorry.
- Don't worry about it Billy, it happens all the time.
Bobby, please determine the binding energy of an object to the planet
or the minimum work necessary to completely remove an object from the planet.
- Okay, sure. So,......
That will be the work done by the force applied
which equals the change in gravitation potential energy of the object
which equals the final gravitational potential energy minus the initial gravitational potential energy.
We can substitute the equation for universal gravitational potential energy in for both the
final and initial values of the gravitational potential energy.
Mass 1 is the object and mass 2 is the earth
Because we are completely removing the object from the earth, the object's final position will be
infinitely far away from the earth.
A negative times a negative is a positive
and the initial distance between the center of the object and the center of the planet
should be the radius of the earth.
Assuming the object starts on the surface of the planet.
Anything divided by infinity is zero.
So, the universal gravitational constant times the mass of the object times the mass of the earth
all divided by the radius of the earth
is the energy which binds the object to the earth.
- So, that is the amount of energy required to completely remove an object from the surface of planet Earth
or the binding energy between the earth and the object.
- [Billy]: Are we not going to substitute in numbers?
- Numbers. So, no. We do not have a numbers dependency.
So, I think we will just leave it right here with variables as our answer.
- Mr. P? - Yes, Bo.
- Would it not have been easier
to just say we are bringing the object to a point infinitely far from the planet
where it will have no gravitational potential energy and no kinetic energy,
so it will end with zero mechanical energy?
If it has a mechanical energy on the surface of the planet of
negative big G times mass 1 times mass 2
all divided by the distance between the centers of mass of the two objects,
then that is how much energy we have to add to it to get it to zero mechanical energy.
So, that would be the binding energy.
- Sure Bo, I think that's a great way of looking at it, so thank you.
Oh, and thank you for learning with me today.
I enjoyed learning with you.