Dimensional Analysis involving power

[d97]

Homework Statement

The power required by a helicopter when hovering depends only upon the vertical thrust (a force) F provided by the blades, their length l, and the mass density of air, ρ. Establish an equation that relates the helicopters power requirement to these three quantities. By what factor is the power requirement increased such that the helicopter can maintain a constant altitude when it takes on a load that doubles its weight?

Homework Equations

[Power]= M*L^2*T^-3, [Density]=[M*L^-3], Length= [L], [Force]= M*L*T^-2
Power= F^a*p^b*L^c[/B]

The Attempt at a Solution

I've done the first part of the question through using dimensional analysis: M*L^2*T^-3=[M*L*T^-2]^a*[M*L^-3]^b*[L]^c. I came up with these equations to get the powers for the RHS: a+b=a, a-3b+c=2 and a=-1. This gave me a=-1, b=2, c=9 this means that Power= F^-1*p^2*L^9, I think that's right?
I'm just unsure on how to do the second part of the question where its asking for a factor of power?
Any help would be much appreciated thank you. [/B]

 

[FactChecker]

Suppose the initial force required to hover is denoted by F₁. Then the force required to hover with twice the weight is 2F₁. Put that into your equations and see what happens to the 2 multiplier.

PS. I didn't try to follow your equations, but it looks like a = -1 means that more lift force requires less power. That seems wrong to me.

 

[kuruman]

Check again your equations. If a+b = a, then b = 0. Also the equation from the exponents of T doesn't look right.

 

[d97]

Check again your equations. If a+b = a, then b = 0. Also the equation from the exponents of T doesn't look right.

Sorry my bad a+b=1, I miss typed

 

[d97]

Suppose the initial force required to hover is denoted by F₁. Then the force required to hover with twice the weight is 2F₁. Put that into your equations and see what happens to the 2 multiplier.

PS. I didn't try to follow your equations, but it looks like a = -1 means that more lift force requires less power. That seems wrong to me.

Oh ok thank you, yeah I think a=-1 is wrong I looked at it again and I think a=3/2 instead. I'll work it out again and I'll put that 2 multiplier into my equation once I've corrected my mistakes.

 

[d97]

Suppose the initial force required to hover is denoted by F₁. Then the force required to hover with twice the weight is 2F₁. Put that into your equations and see what happens to the 2 multiplier.

PS. I didn't try to follow your equations, but it looks like a = -1 means that more lift force requires less power. That seems wrong to me.

I've sorted my mistakes out and I got power= F^3/2*p^-1/2*L^-1, I hope that is right? And put the 2 multiplier into my equation and the power increases by a factor of 2.83, I think that's right?

 

[FactChecker]

I've sorted my mistakes out and I got power= F^3/2*p^-1/2*L^-1, I hope that is right? And put the 2 multiplier into my equation and the power increases by a factor of 2.83, I think that's right?

Sorry. I'm afraid that I don't have the background to check it. I'll have to leave this for people who know the subject matter better than I do.

 

[haruspex]

power= F^3/2*p^-1/2*L^-1

It's right.