Work done moving a satellite into higher orbit

Question

I am confused about the concept of work in relation to conservative and non-conservative forces.

When transferring a satellite from a lower orbit into a higher orbit via a Hohmann transfer, my understanding is that work is initially done to increase the satellite's kinetic energy instantaneously to move it into an elliptical orbit, and then gravity does negative work in converting the satellite's kinetic energy to potential energy (this is simplified and disregards changes in energy in correcting the satellite's orbit into a circular shape).

I learnt that the work done was equal to the change in energy, or you could sum up the work done by the individual forces. I also learnt that work done by conservative forces is equal to change in kinetic energy, whereas work done by non-conservative forces is equal to change in total energy. However in this scenario, both gravity and the non-conservative force used to convert fuel to kinetic energy do work, but if you sum it up it does not equal to the total change in energy which is very confusing.

I'm very lost, please help!

Answer 1

For an object (mass, m) in the gravitational field of the earth (Mass, M) the potential energy is usually expressed as: U = – GMm/r. (This energy is going up from a negative value towards zero as r gets larger.) If the object is in a circular orbit: F = $GMm/(r^2) = m(v^2)/r$. multiplying by r/2 gives the kinetic energy: Gmm/(2r) = $(1/2)mv^2$. Adding these gives the total energy for a circular orbit: E = -GMm/(2r) (which also goes up toward zero). To move to a larger orbit, the rocket adds energy to make the orbit elliptical (which goes higher on the far side) and adds more energy on the far side to make the orbit a (larger) circle.

Answer 2

I learnt that the work done was equal to the change in energy

This statement is too ambiguous. We know that the total work due to all forces (conservative or nonconservative) is equal to the change in the kinetic energy. However, the work done by external forces (usually just taken to be the nonconservative forces) is equal to the change in the total mechanical energy. This is because if start with $$W_\text{total}=\Delta K$$ and then utilize potential energy $$W_\text{cons}=-\Delta U$$ we can split the total work up $$W_\text{total}=W_\text{cons}+W_\text{nc,ext}=-\Delta U+W_\text{nc,ext}$$ which gives us the relationship we want $$W_\text{nc,ext}=\Delta K+\Delta U=\Delta E$$

I also learnt that work done by conservative forces is equal to change in kinetic energy, whereas work done by non-conservative forces is equal to change in total energy.

This is not the case in general.