# Work to move a planet's satellite

• lizzyb
In summary: The kinetic energy is found from the velocity which is found from the equation for the force. So it's not as complicated as it might seem at first.In summary, the work required to move a planet's satellite of mass 2230 kg from a circular orbit of radius 2R to one of radius 3R, where 5.32 X 10^6 m is the radius of the planet, is equal to the change in total energy between the two orbits. This can be calculated using the work-energy principle, which states that the change in total energy is equal to the work done. The potential energy function for a circular orbit is -GMm/r, and the kinetic energy can be found using the
lizzyb

## Homework Statement

Calculate the work required to move a planet's satellite of mass 2230 kg from a circular orbit of radius 2R to one of radius 3R, where 5.32 X 10^6 m is the radius of the planet. The mass of the planet is 3.36 X 10^24 kg.

## Homework Equations

$$U(r) = -\frac{G M m}{r}$$

## The Attempt at a Solution

$$W = \Delta U = U_f - Ui = - \frac{G M_p M_s}{3 R} + \frac{G M_p M_s}{2 R} = - \frac{2 G M_p M_s}{6 R} + \frac{3 G M_p M_s}{6 R} = \frac{G M_p M_s}{6 R}$$

the answer I got (1.5663 X 10^10) was wrong.

Last edited:
You'd have to integrate dU(r) from 2R to 3R; R varies as the work is done.

i came up with the same answer via integration

There will also be a difference in kinetic energy, that is you need to park it in the orbit. Just getting it there is not enough.

Last edited:
I just used

$$\Delta E = E_f - E_i = - \frac{G M_p M_s}{2}(\frac{1}{R_f} - \frac{1}{R_i})$$

and that was the right answer; so the change in total energy is equal to work?

lizzyb said:
I just used

$$\Delta E = E_f - E_i = - \frac{G M_p M_s}{2}(\frac{1}{R_f} - \frac{1}{R_i})$$

and that was the right answer; so the change in total energy is equal to work?

Yes. That is the work-energy principal. The expression you used for the total energy reflects the changes in both potential and kinetic energies between the two orbits. The total energy change is equal to the work done to change the orbit.

How is this equation derived?

andrevdh said:
How is this equation derived?

For a circular orbit

F = mv²/r = GMm/r²

mv² = GMm/r

E = KE + PE = ½mv² - GMm/r = ½GMm/r - GMm/r = -½GMm/r

Thank you. I thought it would be complicated - integration ...?

andrevdh said:
Thank you. I thought it would be complicated - integration ...?

The potential energy function comes from integrating the gravitational force, but that's been done many times so we use the result: PE = -GMm/r

## 1. How does work affect the movement of a planet's satellite?

Work is a measure of the transfer of energy and can have a direct impact on the motion of a planet's satellite. When work is done on a satellite, its kinetic energy and potential energy can change, causing it to speed up or slow down, or move closer or further away from the planet.

## 2. What factors contribute to the amount of work required to move a planet's satellite?

The amount of work needed to move a planet's satellite depends on several factors, including the mass of the satellite, the distance it needs to be moved, and the force applied to it. The greater the mass and distance, and the more force applied, the more work will be required.

## 3. Can the work done on a satellite affect its orbit around the planet?

Yes, the work done on a satellite can change its orbit around the planet. If the work done is in the same direction as the satellite's motion, it can increase its speed and cause it to move to a higher orbit. If the work is in the opposite direction, it can decrease the satellite's speed and cause it to move to a lower orbit.

## 4. How does the direction of the applied force impact the work done on a planet's satellite?

The direction of the applied force has a significant impact on the work done on a planet's satellite. If the force is applied in the same direction as the satellite's motion, more work will be done, resulting in a change in its speed and/or orbit. If the force is applied in the opposite direction, the work done will be negative, reducing the satellite's speed and/or orbit.

## 5. Is work the only factor that affects the movement of a planet's satellite?

No, work is not the only factor that affects the movement of a planet's satellite. Other factors, such as the gravitational pull of the planet and other celestial bodies, can also influence the satellite's motion. These factors can change the satellite's speed, direction, and orbit, thus impacting the amount of work required to move it.

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