- Good morning.
let's determine the total mechanical energy of a satellite.
Bobby, a satellite which is in orbit around a planet has what types of mechanical energy?
Flipping Physics
- Well the satellite is in orbit around a planet,
so the two objects have universal gravitational potential energy.
The satellite is moving so it also has kinetic energy.
There's no type of spring, so no elastic potential energy.
- The equations for that are the negative of the universal gravitational
constant times the mass of the satellite times the mass of the planet all divided by r,
the distance between the center of mass of the planet and the center of mass of the satellite
plus one-half times the mass of the satellite times the velocity of the satellite squared.
- Bo, did you get your haircut?
- Yep. Donated it to Children with Hair Loss.
- That is very kind of you. - Nice.
- You know what Bo, I really like that idea.
Bo, that is definitely a good idea. I will also donate my hair.
This is the mechanical energy of a satellite in orbit around a planet; however,
I would like to simplify the expression to have it not include the velocity of the satellite.
Billy, please draw a free body diagram
of the forces acting on the satellite and solve for the velocity of the satellite.
- Okay.
The only force acting on the satellite is the force of gravity pulling it toward the planet it is orbiting around.
So, we can sum the forces in the in-direction.
Net force in the in-direction equals the force of gravity which equals mass times centripetal acceleration.
Substituting in equations gives us the universal gravitational constant
times the mass of the satellite times the mass of the planet divided by the square of r,
distance between the centers of mass of the satellite and the planet.
Which equals the mass of the satellite times the square of the tangential velocity of the satellite.
Oh, and the tangential velocity of the satellite is just the velocity of the satellite.
Oh, and that is divided by the radius of the orbit.
And the r in the force of gravity equation is also the radius of orbit.
Oh, and...
Everybody brought the mass of the satellite to the party.
- Actually, everybody brought the mass of the satellite divided by the radius to the party.
- Ah. - Yep!
Everybody brought mass!
- And we can solve for the velocity of the satellite in orbit around a planet.
Then going back to the equation for total mechanical energy,
we can substitute in for the square of the velocity of the satellite,
the universal gravitational constant times the mass of the planet divided by the orbital radius.
Factoring out every variable leaves us with negative one plus one half in the parenthesis,
which equals negative one half.
In other words, the total mechanical energy of an orbital object
equals negative one half times the universal gravitational constant times the mass of the satellite
times the mass of the planet all divided by the orbital radius.
That means, the total mechanical energy of a satellite equals half the universal gravitational
potential energy between the satellite and the planet.
- I thought the total mechanical energy was negative?
- It is, remember universal gravitational potential energy is always negative;
the total mechanical energy is just half that negative value.
- Right.
- You know, previously we determined the escape velocity of an object from a planet
to be equal to the square root of two times Big G...
- The universal gravitational constant.
- times the mass of the planet all divided by the distance between the centers of mass of the objects
and the planet.
And, we can relate that to the satellites' velocity.
The escape velocity of an object equals the satellite’s velocity times the square root of two. Why is that?
- Whoa. - Yeah, why is that?
Bo, I do not know why the escape velocity of an object equals the velocity of a satellite
times the square root of two. It is strange.
- Mr. P does not know the answer?
- Thank you very much for learning with me today.
I enjoyed learning with you.