One of the applications of the law of conservation of angular momentum involves a helicopter with a single propeller. A/c the book, a helicopter with one propeller would rotate itself in the opposite direction. However, I am not able to visualise this phenomenon. Can you please explain this, preferably with the help of a figure?
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asked Dec 31, 2020 at 15:57
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1$\begingroup$ youtube.com/watch?v=7FV1vSY30FY or rcgroups.com/forums/… or many others $\endgroup$– BrickCommented Dec 31, 2020 at 16:04
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1$\begingroup$ What book are you referring to? Which page? $\endgroup$– UrbCommented Dec 31, 2020 at 16:43
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$\begingroup$ Why the close vote? This appears to be a legitimate question about a conceptual problem. $\endgroup$– Adrian HowardCommented Dec 31, 2020 at 16:47
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$\begingroup$ @AdrianHoward I didn't vote to close, but the given reason on the current vote was "Needs details or clarity". Personally the request for a figure and the easy access to example online don't strike me as on topic. The OP seems clear on the concept and seems only to be asking for a figure as the question is written. $\endgroup$– BrickCommented Dec 31, 2020 at 17:42
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$\begingroup$ @ Brick Thanks for your response, I, perhaps incorrectly, assumed the poster had a conceptual misunderstanding since he mentions visualization problems. $\endgroup$– Adrian HowardCommented Dec 31, 2020 at 18:38
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4 Answers
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Consider the two phases of motion of the blade of the helicopter. First their angular velocity will increase and then become constant. To produce angular acceleration torque is required. Torque is also required to maintain constant angular velocity otherwise friction will slow it down. Applying anti-clockwise torque would result in clockwise moment generation on the helicopter(except its blades) thus it will start rotating in the opposite direction to the blade.
To escape applying torque analysis on each body we apply angular momentum conservation on the whole system about an axis/point wherever external torque is zero.
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$\begingroup$ As you point out, the friction encountered by the moving blades does exert an external torque, (as does the ground if it is still on the ground); so angular momentum is not conserved in this system. $\endgroup$ Commented Dec 31, 2020 at 19:39
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Here is a simple analogy. Assume you are standing upright, stationary on ice (frictionless surface). The external forces (gravity and constraint from surface) provide no torque about your center of mass (CM), so the angular momentum about your CM is constant and zero. If you swing your arms to the left your body rotates to the right to keep the angular momentum zero.
For the helicopter the propeller is the "arms" and the body of the helicopter is the "body".
Hope this helps.
answered Dec 31, 2020 at 16:39
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As the helicopter applies torque to its top rotor, the rotor is applying an equal and opposite torque to the helicopter. The helicopter and the top rotor would be spinning in opposite directions without the tail rotor to push against this equal and opposite torque applied by the top rotor to the helicopter.
answered Dec 31, 2020 at 16:40
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Conservation of angular momentum. Since no external torques are applied to the helicopter (the forces causing the torques that spin the rotor are internal), the total angular momentum must stay constant, and equal to zero (since initially there is no angular speed). In other words, the following must be true $$I_\mathrm{blade}\omega_\mathrm{blade} + I_\mathrm{heli}\omega_\mathrm{heli}=0$$
Therefore, if $\omega_\mathrm{blade}$ is positive, then $\omega_\mathrm{heli}$ will be negative, i.e. the helicopter must spin in the opposite direction.
answered Dec 31, 2020 at 16:48
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$\begingroup$ Note that the blades can also exchange momentum with the air which you can not neglect here (if we neglect the air the helicopter would never leave the ground). The general picture is still true but I wouldn't call this a closed system. $\endgroup$ Commented Apr 18, 2022 at 19:39