Calculating the center of gravity (hard version)

[mathguy2]

Homework Statement

A Cessna C-172M has empty weight 1350 lbs. When it is weighed, the load on each of the main wheels is 474 lbs. The load on the nose wheel is 402 lbs. Where is the center of gravity of the empty Cessna?

Ok so the question assumes you know what a Cessna looks like, but it's like this: the nose wheel bears 402 lbs. The main wheels bear 948lbs (2 wheels so 474 x 2 = 948). 948 + 402 = 1350, so it's correct.

Homework Equations

I believe this is the equation for translating 2 weights into 1 weight (the center of gravity being the 1 weight)

L₁ = (L x W₂) / W

L₂ = (L x W₁) / W

The Attempt at a Solution

I apologize, but this is really confusing me. The reason I'm struggling is that I don't have any lengths, which I guess might mean that this is a 2 stage problem, but I can't seem to figure out how to even get started. If you look at my recent posts, you can see me working through similar problems, but this one is really throwing a curve at me.

I keep solving until L₁ = 948 / 1350 and
L₂ = 402 / 1350

but from here (even if this a correct first step), I have no clue how to go. Feedback would be really appreciated because I've been staring at this for over an hour (sadly).

 

[gneill]

Two options:
1. Go crazy and hunt down the dimensions of the plane to find the actual distance between the nose wheel and main wheels, or
2. Work symbolically by declaring the distance to be D and present an expression including D.

 

[mathguy2]

So there's no way to get a non-symbolic answer?

 

[gneill]

So there's no way to get a non-symbolic answer?

Not without coming up with actual dimensions.

 

[mathguy2]

Ugh,

Ok so doing it symbolically, where L = D for (Distance from main wheels to nose wheel)

I get...

L₁ = (D x 948) / 1350

L₂ = (D x 402) / 1350

but to where from here?

 

[gneill]

Since you don't have any actual dimensions a practical approach is to take one of the given landmarks as a reference point and express the center of mass in terms of the fraction of the distance between it and the other landmark. If you take the nose wheel as the reference point then the weight measured there is zero distance from it, but the weight at the main wheels is at distance D from it. You can write a single expression for the location of the center of mass using the individual weights and distances from the chosen reference point. In general, if you have a set of i = 1..n weights and their distances from the point of reference: w_i and d_i then the center of mass will be given by
$$d_{cm} = \frac{\displaystyle\sum_{i = 1}^{n} w_i d_i}{\displaystyle\sum_{i = 1}^{n} w_i}$$