Shear modulus, finding length of board

[DracoMalfoy]

Homework Statement

A diver stands (80kg) at the end of a diving board, causing the board to dip 10cm. The board has a width of 34cm and a height of 6cm. If the shear modulus of the board is 2x10^6pa, what is the length of the board? (Ans: 5.2m)

Homework Equations

[/B]
F/A=S⋅Δx/h

The Attempt at a Solution

F=80(9.8)= 784N
A=.34(.06)=.0204
S=2x106
h=.06

Im not really sure how to find the change in x. I am not even sure if this is the correct equation to use.

 

[gneill]

F/A=S⋅Δx/h

That should be

$\frac{F}{A} = S \frac{\Delta x}{L}$

Take a look at the Wikipedia article on Shear modulus and the diagram at the top of the page.

https://en.wikipedia.org/wiki/Shear_modulus

 

[haruspex]

I don't understand how shear modulus comes into it. The board dips because it bends, and that is to do with elastic modulus, not shear modulus.

 

[Chestermiller]

I don't understand how shear modulus comes into it. The board dips because it bends, and that is to do with elastic modulus, not shear modulus.

I agree. The equation for bending a cantilever beam should be used. The elastic modulus can be determined from the shear modulus if the Poisson ratio is known. In lieu of data, a value of 0.3 should be adequate.

 

[gneill]

I don't understand how shear modulus comes into it. The board dips because it bends, and that is to do with elastic modulus, not shear modulus.

I had similar trepidations about the question as posed. I wasn't sure if the intent was to work the problem as though it were a shear modulus problem or to get more "creative". The problems that the OP has posted previously did not require more insight than rote application of the relevant equations, so I figured "introductory course".

 

[Chestermiller]

I had similar trepidations about the question as posed. I wasn't sure if the intent was to work the problem as though it were a shear modulus problem or to get more "creative". The problems that the OP has posted previously did not require more insight than rote application of the relevant equations, so I figured "introductory course".

The problem statement itself does not present a correct mechanistic representation of what is actually involved physically. It misleads the student.

 

[gneill]

The problem statement itself does not present a correct mechanistic representation of what is actually involved physically. It misleads the student.

I agree 100%. It's a very poorly formulated question.

So, do we help with the shear modulus calculation (wrong, but what the question asks for) and then provide a correct interpretation/solution? I will go by your judgement here.

 

[Chestermiller]

I agree 100%. It's a very poorly formulated question.

So, do we help with the shear modulus calculation (wrong, but what the question asks for) and then provide a correct interpretation/solution? I will go by your judgement here.

Yes. Thanks. I’ll get back to this later after performing my duties handing out Halloween candy.

 

[gneill]

Yes. Thanks. I’ll get back to this later after performing my duties handing out Halloween candy.

Yeah, similarly deluged here

 

[haruspex]

The elastic modulus can be determined from the shear modulus if the Poisson ratio is known. In lieu of data, a value of 0.3 should be adequate.

To get the given answer treating it as a cantilever, I need the elastic modulus to be 3/5 of the given shear modulus. I think that would require Poisson's ratio to be negative.

 

[Chestermiller]

@DracoMalfoy : Your teacher has done you a bit of a disservice here by assigning you this problem with this particular problem statement. The implication of the problem statement is that the deformation of the diving board occurs primarily by shear. This would result in a diving board concur which is a slanted straight line from one end to the other. But this is not how a diving board behaves in the real world. In the real world, the diving board contour is a curved arc, and the mechanism of deformation is not shear; it is bending. Anyone who has ever seen a diving board with a person standing at its end knows this. All an assignment and problem statement like this serves to do is to confuse the student, not only for this problem, but for future problems in which he is learning about the actual deformation of beam bending.

 

[Chestermiller]

To get the given answer treating it as a cantilever, I need the elastic modulus to be 3/5 of the given shear modulus. I think that would require Poisson's ratio to be negative.

That's because they assumed the incorrect bending mechanism. The true answer will result in a different length.

 

[haruspex]

That's because they assumed the incorrect bending mechanism. The true answer will result in a different length.

Yes, I was just establishing that it was neither a matter of the given modulus having been wrongly titled, nor of the student being expected to approximate the one modulus from the other.