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Forces on a Helicopter

Helicopters are fascinating creatures. One of their distinguishing features is their ability to hover in place. This is essentially accomplished through a torque and force balance. For the forces we must look at Newton’s second law of motion:

Newton's Second Law

Notice the arrows on top of the force and acceleration terms. These mean we’re dealing with force vectors, and an acceleration vector. These vectors have three components because helicopters live in three dimensions. For a helicopter to remain stationary the acceleration must equal zero, i.e., all three components of the acceleration vector must be zero. By setting the acceleration vector to zero in our equation, we see that the left side of the equation must be zero as well. Thus, the only way to achieve zero acceleration is if the sum of the forces add up to zero.

For the torques we turn to Euler’s (simplified) equation of motion:

Euler's Equation of Motion

It looks suspiciously similar to the first force equation above. In essence, torques are exactly the same as forces except that they deal with rotation instead of translation. If the sum of the torques aren’t zero it will cause the helicopter to spin around. Thus, for a helicopter to remain stationary both the sum of the forces, and the sum of the torques must be zero.

Gravity is the only force that acts on a helicopter when it’s hovering, so all we have to do is supply a force that counteracts gravity and we’re set. This is the job of the main blades. However, these spinning blades introduce a new torque on the helicopter, which throws off the torque balance and causes the helicopter to spin in the opposite direction of the main blade direction. This new torque is balanced with the tail rotor (on a conventional helicopter) that pushes the tail of the helicopter in the opposite direction of the spin. Thus the torques are balanced once again.  But we’re not finished yet. Similarly to how the main blades caused a torque as well as a force, by using a tail rotor to balance the torque, another force has been introduced that pushes the helicopter to one side or the other (depending on which way the tail rotor is facing). Add another propeller you say? Well it turns out that this force can be balanced simply by tilting the helicopter so that the force from the main blades acts slightly to one side. This balances the small force caused by the tail rotor. Finally all the forces and torques are balanced and we have a stationary helicopter hovering in the air. The next time you see a helicopter hovering stationary, see if you can see how it leans very slightly to one side to counteract the force from the tail rotor. This can be seen in the following video:

Image from R/C Airplane World.