How can I calculate the torque on a body resulting from a spinning propeller? If a quadcopter has all of its propellers spinning on the same direction, the body of the quad will obviously start spinning in the another direction.
I know how to calculate the torque acting on a object but in this case  the torque is caused by the propeller spinning on a constrained axis that is not the same as the axis of rotation of the body. How can the three dimensional torque vector acting on the body (the black rectangle in the figure with center of mass at CM) caused by the accelerating propeller (blue and spinning counterclockwise) be calculated?

 A: If the blade was spinning in a vacuum (and there was no friction and no motor), there would be no torque. The blade could spin by itself forever. 
Torque comes from spinning in air and pushing air around. Air is viscous. It resists the motion of the blades. It exerts a torque on the blades. The torque on all 4 blades is the torque on the quadcopter.
Figuring out the torque on a blade requires knowing the shape of the blade. It is possible, but not trivial. You would want software to help. 
There is an easier way. The motor has to exert an equal torque on the blades to keep them moving. If you know the power of the motor and the angular velocity of the blades, you can use 
$P = \tau\omega$
As for how the torque exerted at the end of a stick affects the stick, you haven't drawn everything that is important. If you twisted a free propeller as shown, it would spin without exerting a torque on the stick. But there is a motor at the center and a belt connecting it to the propeller. Air resistance at the blades is transmitted to the belt, which is transmitted to the motor, which is bolted to the quadcopter.
