Largest mass for Ultimate Tensile Strength

[AJKing]

Vibrations and Waves, A.P. French, questions 3-9
Please /do not/ provide full solutions. Just suggestions.

Homework Statement

A solid steel ball is hung at the bottom of a steel wire of length 2m and radius 1mm, the ultimate strength of steel is 1.1E9 N/m². What are the radius and mass of the biggest ball the wire can bear?

Homework Equations

$Y = \frac{\sigma}{\epsilon}$ (line)

$U = \frac{\sigma_u}{\epsilon_u}$ (point below line)

$m = \frac{4}{3}\pi*r^3 \rho$

$\rho \approx 7850 Kg/m^3$ (Internet)

The Attempt at a Solution

For the sake of sanity, I arrange eq.2 by knowns and unknowns.

$\frac{U*A}{g*l_0} = \frac{m}{\Delta l}$

I must find Δ l

I consider two points on the Y line:

$Y = \frac{\sigma_u}{\epsilon_1}$

$Y = \frac{\sigma_2}{\epsilon_u}$

both can be written as

$Y = C\frac{\sigma_u}{\epsilon_u} = C U$

$C = Y/U$

Then I stop and think about how this will lead me in circles.

------

I have a suspicion that the solution to my problem requires differential equations. Anyone second that notion?

 

[SteamKing]

DEs not required. Do you know what Y represents?

 

[SteamKing]

You know the ultimate stress the wire is capable of sustaining. You also know the size of the wire. Can you use this information to work back and figure out the amount of mass required to produce this stress? Draw a free body diagram if it helps you to visualize the situation.

 

[AJKing]

You know the ultimate stress the wire is capable of sustaining. You also know the size of the wire. Can you use this information to work back and figure out the amount of mass required to produce this stress? Draw a free body diagram if it helps you to visualize the situation.

Y is available in the text.

[strike]The ultimate stress σ_u. Do I really know this without knowing the ultimate strain ε_u?[/strike]

Have I been wrong this entire time? Is ultimate strength just the stress associated with failure?
Suggesting the following is incorrect:

$U = \frac{\sigma_u}{\epsilon_u}$

 

[AJKing]

Holy hell.

Yes. I was incorrect about ultimate strength!
Break through in understanding here folks.

Ultimate strength is the stress of failure.

Question solved correctly.

Our discussion in the other forum makes more sense now SteamKing - cheers :).