I would be grateful if you might help me with a difficult problem in mechanics that I have tried to solve for a long time without success:
How much power P [W] is needed to force a satellite (modeled as a point particle) to move in a perfect circular orbit with radius r [m]? The mass of the satellite is m [kg] and its tangential speed v [m/s] is constant.
- It is assumed that NO GRAVITY affects the satellite!
- The mass loss due to fuel consumption is neglected.
- The power is assumed to act under ideal conditions, i.e. there is no energy loss due to friction or the choice of engine.
- Relativistic effects are neglected.
I have received many answers saying that P = 0 as a consequence of the definition of mechanical work. But this cannot be true since no gravity affects the satellite. Energy is needed to create a force that changes a linear path in space and a circular orbit is one such motion. In absence of gravity no particle with mass will loop in a circle without any energy.
What you have been told before is correct. No work is done by the force causing the satellite to move in orbit. Since no work is done, we can ask whether the energy is conserved. Since there is no gravity, the only energy to think about is the kinetic energy of the satellite and since the speed is the same always, so is the kinetic energy. Or, look at it this way: suppose we are in otherwise empty space but that there is a fixed point to which we can tie a rope, the other end of which we tie to the satellite which we give a shove so that it moves in a circle around the center. Does the string need to supply any energy? Do you need to connect a machine of some sort to the rope? Your basic premise, "Energy is needed to change a linear path in space", is simply wrong; a force is needed, but not energy.