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Dimensional analysis check

  • #1
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Homework Statement



An flying object of mass M experiences a lift force and a drag force dependent on the it's velocity:

[tex] F_{l} \propto Wv^{2} [/tex]
[tex] F_{d} \propto Av^{2} [/tex]

Where W and A are wing surface area and "cross-sectional" area.

To sustain flight the object must fly at a minimum velocity which scales as [itex]M^{k} [/itex] where k is a constant.

What is k?

Homework Equations



For the object to remain in flight it must produce a lift force equal or greater than it's weight i.e.

[tex] Mg = F_{l(min)} \propto Wv^{2} [/tex]



The Attempt at a Solution




I have found an answer but I'm not too happy with how I got there. Basically, I subbed the scaling factor for minimum velocity into the above equation:

[tex] Mg \propto W(M^{k})^{2} \propto WM^{2k} [/tex]

As the left side only has mass to the power 1 and the right has mass to the power 2k we can get:

[tex] 2k=1 [/tex]

So k = 1/2 or [itex]v_{l(min)} \propto \sqrt{M} [/itex].

This results makes some sense to me physically, but I'm not happy with my approach as it seems a bit simplistic.
 
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Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

Homework Statement



An flying object of mass M experiences a lift force and a drag force dependent on the it's velocity:

[tex] F_{l} \propto Wv^{2} [/tex]
[tex] F_{d} \propto Av^{2} [/tex]

Where W and A are wing surface area and "cross-sectional" area.

To sustain flight the object must fly at a minimum velocity which scales as [itex]M^{k} [/itex] where k is a constant.

What is k?

Homework Equations



For the object to remain in flight it must produce a lift force equal or greater than it's weight i.e.

[tex] Mg = F_{l(min)} \propto Wv^{2} [/tex]



The Attempt at a Solution




I have found an answer but I'm not too happy with how I got there. Basically, I subbed the scaling factor for minimum velocity into the above equation:

[tex] Mg \propto W(M^{k})^{2} \propto WM^{2k} [/tex]

As the left side only has mass to the power 1 and the right has mass to the power 2k we can get:

[tex] 2k=1 [/tex]

So k = 1/2 or [itex]v_{l(min)} \propto \sqrt{M} [/itex].

This results makes some sense to me physically, but I'm not happy with my approach as it seems a bit simplistic.
Seems fine to me. If you increase the mass by a factor of ##2## you need to increase the velocity by a factor of ##\sqrt{2}##, all else being fixed. What seems simplistic about it to you?
 
  • #3
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
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Maybe you are supposed to assume that the object is to be scaled linearly at constant density. Thus W and A will also change.
 

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