Can Dimensional Analysis Solve the Helicopter Hovering Dilemma?

[Mircea Golumba]

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.

 

[Andy Resnick]

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.

 

[Mircea Golumba]

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.

 

[Mircea Golumba]

Later edit: sorry, my formula above is wrong, please consider this instead:
http://www.educatiarutiera.ro/wp-content/uploads/2017/03/sol592.png

 

[anorlunda]

Thanks for sharing that. I never heard of Buckingham π theorem before. It is most interesting and clever.

 

[Dr.D]

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.