ORIGINAL: low@slow
Rupert, load on a prop goes up by the square of the speed because you are increasing the static thrust, as well the velocity it is being generated at. So to double the rpm of a given prop, you would need four times the power. Also, drag, i.e. friction, goes up by the square of the speed as well.
Lyn,
You have to get your terms straight...
Are you referring to 'power' as in horsepower, which is a unit of output, or as in torque, which is force?
If you are referring to torque, you are correct.
Since air resistance and thus prop drag, rises as a function of the air-speed squared, spinning a prop twice as fast would require four times the torque.
If you are referring to horsepower, than you are wrong, since eight (8) times the horsepower would be required to spin the same prop twice as fast... The required torque is four times greater and this force must be applied over twice the distance per unit of time, so the required work per time unit is eight times greater at double the RPM.
At three times the RPM it is 27 times greater and at four times the RPM, 96 times greater.