Help with elasticity/stretching&compression

[Nellen2222]

Homework Statement

A Force stretches a wire by 1.0mm A second wire of the same material has the same cross section and twice the length. How far will it be stretched by the same force?

b) A third wire of the same material has the same length and twice the diameter as the first. How far will it be stretched by the same force?

Homework Equations

F = y(delta)L/L0*A Youngs modulus

The Attempt at a Solution

1.0mm = 0.01m

I believe you have to compare the original youngs mudulus with one that is slightly different... I need some guidelines on how to solve this mathematically please. Also, on a side note, is there a differnence between Youngs modulus/shears modulus/bulk modulus? or all they all aprt of the same idea with different concepts, ie(one uses volume)

Thanks

 

[CWatters]

I believe you have to compare the original youngs mudulus with one that is slightly different...

No the material is the same so the youngs modulus should be the same.

Try this approach..

F = -EA₀ΔL/L₀

E is the Young's modulus (modulus of elasticity)
F is the force exerted on an object under tension;
A₀ is the original cross-sectional area through which the force is applied;
ΔL is the amount by which the length of the object changes;
L₀ is the original length of the object.

So write two versions of the above one for the short wire (subscript s) and one for the long wire (subscript L). The youngs modulus, the cross sectional area and the force are the same for both lengths of wire..

F = -EA₀ΔL_s/L_s

F = -EA₀ΔL_L/L_L

Since the force is the same you can equate the two...

-EA₀ΔL_s/L_s = -EA₀ΔL_L/L_L

The youngs modulus and the cross sectional area cancel leaving..

ΔL_s/L_s = ΔL_L/L_L

Then rearrange to give...

ΔL_L/ΔL_s= L_L/L_s

In other words the amount of stretch (ΔL) increases in the same ratio as the length (L) increases. eg Double the length and the amount of stretch is also doubled (all else being equal).

If
L_L/L_s = 2
then
ΔL_L/ΔL_s = 2 Part 2 can be tackled in a similar way. Write two equations with different "A". Note that in this case the length L, the force F and the youngs modulus is the same for both.