How Does the Energy Principle Help Calculate Work Needed to Move a Satellite?

[Kibbel]

Homework Statement

In certain cases, using both the momentum principle and energy principle to analyze a system is useful, as they each can reveal different information. You will use the both momentum principle and the energy principle in this problem.

A satellite of mass 7000 kg orbits the Earth in a circular orbit of radius of 8.6 106 m (this is above the Earth's atmosphere).The mass of the Earth is 6.0 1024 kg.
What is the magnitude of the gravitational force on the satellite due to the earth?
F = 37,714 N

Using the momentum principle, find the speed of the satellite in orbit.
v = 6906.93 m/s

Using the energy principle, find the minimum amount of work needed to move the satellite from this orbit to a location very far away from the Earth. (You can think of this energy as being supplied by work due to something outside of the system of the Earth and the satellite.)
work = ?

Homework Equations

for this problem, I seriously don't know, I was working with escape speed (mv^2)/2 + (-Gmm/R) = 0, but that includes the planet as the system, which I am not sure I should

The Attempt at a Solution

help, like my attempt probably was wrong. and would lead off track

 

[The legend]

it's ok if it's wrong, just post the attempt and it can be corrected.

 

[Kibbel]

okay here's my attempt

I said that (V being the initial velocity) (mv^2)/2 + (-Gmm/R) = 0 would give the initial velocity to escape the pull of gravity. Which in turn means that moving Gmm/r to the other side would give us the necessary kinetic energy to escape Earth's gravitational pull. (or the energy that gravity is applying)

So if I subtract the initial kinetic energy the spaceship has from the initial kinetic energy required to hit escape velocity, would that be correct?

 

[Kibbel]

couldn't get it :/ bummer

 

[rwisz]

I would like to clarify that the energy principle and work are not the same thing. The energy principle is a fundamental concept in physics that states that energy cannot be created or destroyed, only transferred or transformed. Work, on the other hand, is a measure of the energy transferred to or from a system by a force acting on it.

In this problem, we can use both the momentum principle and the energy principle to analyze the motion of the satellite. The momentum principle tells us that the net force on the satellite is equal to the rate of change of its momentum. In this case, the only force acting on the satellite is the gravitational force from the Earth, which is given by the equation F = GmM/r^2, where G is the gravitational constant, m is the mass of the satellite, M is the mass of the Earth, and r is the radius of the orbit. Therefore, we can calculate the speed of the satellite using the equation F = m(v^2)/r, where v is the speed of the satellite.

To find the minimum amount of work needed to move the satellite from its current orbit to a location very far away from the Earth, we can use the energy principle. The energy of the satellite in its current orbit is given by the equation E = (mv^2)/2 - (GmM)/r. To move the satellite to a location very far away from the Earth, we need to increase its potential energy by a certain amount. This can be done by applying a force to the satellite over a certain distance, which is equivalent to doing work on the satellite. Therefore, the minimum amount of work needed to move the satellite from its current orbit to a location very far away from the Earth is equal to the change in its potential energy, which is given by the equation W = (GmM)/r - (GmM)/r2, where r2 is the distance to the new location.

In summary, the energy principle and work are closely related but not the same thing. In this problem, we can use the momentum principle to find the speed of the satellite in orbit and the energy principle to calculate the minimum amount of work needed to move the satellite to a new location. These principles each provide valuable information about the system and can be used together to gain a more complete understanding of the satellite's motion.