# Equilibrium/ Young's Moduli

1. Dec 3, 2006

### cd80187

In Fig. 12-49, a lead brick rests horizontally on cylinders A and B. The areas of the top faces of the cylinders are related by AA = 2.4 AB; the Young's moduli of the cylinders are related by EA = 2.2 EB. The cylinders had identical lengths before the brick was placed on them. What fraction of the brick's mass is supported (a) by cylinder A and (b) by cylinder B? The horizontal distances between the center of mass of the brick and the centerlines of the cylinders are dA for cylinder A and dB for cylinder B. (c) What is the ratio dA /dB?

So since I don't know how to post the picture, i'll just describe it. There is a silver block placed on top of cylinder A on the left and cylinder B on the right. A has a larger diameter than B. The com of mass is labeled in between cylinder A and B, but a little bit closer to B. Two lines are drawn, one connecting the center of cylinder A to the center of mass of the brick, labelled d subscript A. There is another line drawn from cylinder B to the bricks center of mass labelled d subscript B.

Now for this problem, I do not even know where to start. I think that I am supposed to use the equation F/A = E ( change in L/L). I have no clue how to use this equation though, but it is the only one that is relevant to the problem. So also, could you explain this equation, because I have no clue how to use. Thank you for the help

2. Dec 3, 2006

### PhanthomJay

The key here is that the brick rests horizontally on the cylinders. Therefore, each cylinder must deform by the same amount. Find the deformation of each cylinder using the axial deflection formula, and set them equal. Then use the static equilibrium formulae for sum of torques and forces = 0.

3. Jun 22, 2009

### turandorf

I'm having trouble with this one too. I know that the change in length of one equals the change in length of the other which means y1(F1/A1)L=y2(F2/A2) where y=Young's mod, F equals the force of gravity, L equals the total length of the brick, and A equals the area. So using the given numbers you get EA(F1/AA)L=2.2EB(F2/2.4AB)L. I'm just clear how, from these equations, do you get the fraction of weight each is supporting?
I also know you need to use the dA and dB (distances from the center of mass) such that (dA+dB)F2 and F1+F2=mg. Any help would be appreciated! Thanks!

4. Jun 22, 2009

### ideasrule

"y1(F1/A1)L=y2(F2/A2) where y=Young's mod, F equals the force of gravity, L equals the total length of the brick"

You're setting the change in length of the cylinders equal; why does the length of the brick matter?

"EA(F1/AA)L=2.2EB(F2/2.4AB)L"

Despite the mistake you made in your previous equation, this one is correct. The "L"'s cancel out, and you can calculate F1/F2. That's the ratio of the weight the first cylinder supports to the weight the second supports. Do you know how to convert a ratio into a fraction?

5. Jun 22, 2009

### turandorf

So you would get EA(F1/AA)=2.2EB(F2/2.4AB). When I solve for F1, I get (.2EB(F2/2.4AB)/EA)*AA and F2=((EA(F1/AA))/(2.2EB))*2.4AB. Not sure where to go from here. And I'm not sure how to convert a ration to a fraction. Thanks for helping btw.

6. Jun 22, 2009

### PhanthomJay

I find these terms EA, EB, AA, AB very confusing the way they are written. Anyway, the deflections of both cylinders are equal. That implies

$$F_AL/A_AE_A = F_BL/A_BE_B$$

And since $$A_A = 2.4A_B$$ and $$E_A = 2.2E_B$$, then

$$F_AL/((2.4A_B)(2.2E_B)) = F_BL/A_BE_B$$

The $$L$$,$$A_B$$ and $$E_B$$ cancel, and you are left with

$$F_A/((2.4)(2.2)) = F_B$$, or $$F_A = 5.28F_B$$.

Now continue....

7. Jun 22, 2009

### turandorf

Thx!! That really helps! But how do you convert 1:5.28 to a fraction part of the whole? Haven't had to do this in a while. :P

8. Jun 22, 2009

### turandorf

Oh nvrmind I got it! Thanks anyway!

9. Jan 13, 2012

### Squeeg

I haven't either, could someone give me a hand with this step?

I also have another step to this problem: The horizontal distances between the center of mass of the brick and the centerlines of the cylinders are dA for cylinder A and dB for cylinder B. What is the ratio dA/dB?