Question 2.6.5: A helicopter rotor whose length is R = 5.0 m pushes air down...
A helicopter rotor whose length is R = 5.0 m pushes air downwards with a speed v. Assuming that the density of air is constant at ρ = 1.20 kg m^{-3} and the mass of the helicopter is 1200 kg, find v. You may assume that the rotor forces the air in a circle of radius R (spanned by the rotor) to move with the downward speed v.
The momentum of the air under the rotor is mv, where m is the mass of air in a circle of radius 5.0 m. In time Δt the mass is enclosed in a cylinder of radius R and height vΔt. Thus, the momentum of this mass is \rho \pi R^{2} v^{2} \Delta t and its rate of change is \rho \pi R^{2} v^{2}. This is the upward force on the helicopter, which must equal the helicopter’s weight of 12000 N. So
\rho \pi R^{2} v^{2}=M g\Rightarrow v=\sqrt{\frac{M g}{\rho \pi R^{2}}}
Thus, v = 11 m s^{-1}.
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