Work of a spaceship in circular motion Say a spaceship is traveling though space in a uniform circular motion.
It's not orbiting any planet, it just flies in circles in an empty space.
The only force working on the spaceship would be the centripetal force caused by the ship's engine.
Thus, the work would be $0$, as the force would always be perpendicular to the ship's path.
But that sounds counterintuitive to me, it would seem that the spaceship must do some work, otherwise it would just float in a straight line.
Can anyone point out the error in my reasoning?
 A: You are right in saying that the centripetal force doesn't do any work, in fact the kinetic energy of the system doesn't increase as the absolute value of the velocity $|\vec{v}|$ stays constant.
I guess what you find counter intuitive is that the spaceship has to burn some fuel to keep rotating, so where does this energy go? Simply it is in the fuel.
To keep the rotation, the spaceship will need to keep ejecting mass, in particular if the centripetal force you need is $\vec{F}$, then from Newton's law, every $dt$ you need the change in momentum $\vec{F}dt= md\vec{v}$, where  I neglected the loss of mass of the spaceship for simplicity. Since the system is isolated the momentum has to be conserved, therefore for every $dt$ you need to eject some fuel carrying that much momentum (with vector pointing outwards). This ejected fuel will also carry the energy we were looking for.
A: Without a force the spaceship would be floating in straight line at constant velocity. The reason why work is zero and the object is accelerated comes from its very definition.
From the second Newton's law:
$$\mathbf F = m\frac{d\mathbf v}{dt}$$
Making a dot product with an infinitesimal displacement:
$$\mathbf {F.dr} = m\frac{d\mathbf v}{dt}\mathbf {.dr} = m\mathbf {v.}d\mathbf v = d\left(\frac{1}{2}m\mathbf {v.v}\right) = d\left(\frac{1}{2}m|v|^2\right)$$
Because $\mathbf {F.dr} = dw$ by definition, it is zero if there is no change in the modulus of the velocity, even if the velocity vector change.
A: This is  a continuation  of Andrea's and Claudio's answers:
From this link

Work refers to an activity involving a force and movement in the direction of the force. A force of 20 newtons pushing an object 5 meters in the direction of the force does 100 joules of work.


Energy is the capacity for doing work. You must have energy to accomplish work - it is like the "currency" for performing work. To do 100 joules of work, you must expend 100 joules of energy.

italics mine
Energy is a conserved quantity, work is not a conserved quantity, as its definition relies on the vector direction of the force, as Claudio states.
So energy is conserved as discussed in the answer by Andrea, but work in this particular problem is zero if the loss of mass is ignored.
If the loss of mass is not ignored, the magnitude of F should change because of the diminution of mass, so as to stay in the same r radius circle,and then there is work done.
A: Thanks everyone for answering!
I find you explanations compelling, I would like to just add one more example in which it's much more intuitive for me to understand that no work is done in circular motion:
Say that our spaceship is attached to a string, and that the other end of that string is attached to the center of the hypothetical circle.
Once the shuttle is in motion, the string will keep it going in circles forever, no engines required, no work done!
