Can Dimensional Analysis Solve the Helicopter Hovering Dilemma?

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Discussion Overview

The discussion revolves around a problem from "200 Puzzling Physics Problems" regarding the power output required for a helicopter to hover, particularly focusing on how dimensional analysis can be applied to determine the rotor speed of a smaller helicopter with half the linear dimensions of the original. Participants explore the implications of using dimensional analysis and the Buckingham π theorem in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the reasoning behind using power output and suggests exploring rotor speed instead, proposing a dimensional analysis approach.
  • Another participant argues that motor power is proportional to output torque multiplied by rotation frequency, implying a more complex relationship than initially considered.
  • A later reply introduces the Buckingham π theorem, indicating that it may provide additional insights into the problem, although it was not utilized in the original formula presented.
  • One participant acknowledges a mistake in their earlier formula and provides a corrected version.
  • Another participant expresses interest in the Buckingham π theorem, noting its relevance and historical significance in dimensional analysis.

Areas of Agreement / Disagreement

Participants express differing views on the application of dimensional analysis and the relationships between power, torque, and rotor speed. There is no consensus on the correct approach or the implications of the Buckingham π theorem in this specific problem.

Contextual Notes

Some participants mention limitations in their understanding of dimensional analysis and the Buckingham π theorem, indicating a need for further study and clarification. There are unresolved mathematical steps and assumptions regarding the relationships between the variables involved.

Mircea Golumba
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Hi, this problem is bugging me for several days now. In ”200 Puzzling Physics Problems”, Gnadik, Honyek, Riley there is this Problem 59:
A helicopter can hover when the power output of its engine is P. A second helicopter is an exact copy of the first one, but its linear dimensions are half those of the original. What power output is needed to enable this second helicopter to hover?

The solution to this is an ingenious resort to dimensional analysis, as follows:
http://www.educatiarutiera.ro/wp-content/uploads/2017/03/sol59.png
Now, I don't have a problem with this. I wonder however, what if we express, instead of the power needed to hover, the rotor speed, or frequency, that has the dimension s-1.
this would lead to α=1/2, β=-1/2 and γ=-δ=k. So, we'll get for the rotor speed s, needed for the helicopter to hoover:
http://www.educatiarutiera.ro/wp-content/uploads/2017/03/sol591.png
Please let me know wether my reasoning is correct. I'm not familiar with dimensional analysis and I'd really want to introduce this problem to some of my students. Thank you.
 
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Mircea Golumba said:
Hi, this problem is bugging me for several days now. In ”200 Puzzling Physics Problems”, Gnadik, Honyek, Riley there is this Problem 59:

The solution to this is an ingenious resort to dimensional analysis, as follows:
Now, I don't have a problem with this. I wonder however, what if we express, instead of the power needed to hover, the rotor speed, or frequency, that has the dimension s-1.
this would lead to α=1/2, β=-1/2 and γ=-δ=k. So, we'll get for the rotor speed s, needed for the helicopter to hoover:
http://www.educatiarutiera.ro/wp-content/uploads/2017/03/sol591.png
Please let me know wether my reasoning is correct. I'm not familiar with dimensional analysis and I'd really want to introduce this problem to some of my students. Thank you.

Awesome problem! But I think the motor power is proportional to the output torque * rotation frequency, not simply the rotation frequency. If I did the problem correctly, if the output torque is held constant, the rotor frequency of the smaller helicopter is faster by 2^(7/2). That said, it's early and the caffeine hasn't kicked in... but at least it agrees with observation: small birds and insects move their wings much faster than larger birds.
 
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Thanks for reply but oops, I just found about something called ”Buckingham π theorem” - details - that is apparently disregarded by Gnadik's formula. I intend to study more but I could really make use of somebody more experienced's insight.
 
Later edit: sorry, my formula above is wrong, please consider this instead:
http://www.educatiarutiera.ro/wp-content/uploads/2017/03/sol592.png
 
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Thanks for sharing that. I never heard of Buckingham π theorem before. It is most interesting and clever.
 
The Buckingham Pi Theorem is well known in dimensional analysis. I encountered it in graduate school 50 years ago. A bit of Internet research will no doubt turn up more info on it.
 

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