# Tensile strain produced on two thin rods of different lengths

Suppose there are two thin rods $$Y$$ and $$Z$$ with length $$L_1$$ and $$L_2$$ respectively. $$L_2$$ has larger magnitude than $$L_1$$. Both rods have same density $$p$$, cross sectional area $$A$$, Young's Modulus $$E$$ and force applied perpendicularly to the cross sectional area $$F$$.

For simple compression or tension we have equation: $${F\over A}= E {\Delta L\over L}$$ that relates tensile stress and strain.

We can also write this equation as: $${FL_1\over AE}={\Delta L_1}$$ and $${FL_2\over AE}= {\Delta L_2}$$ for rod with length $$L_1$$ and $$L_2$$ respectively.

We can notice from above equations that same force $$F$$ causes more change in length ($$\Delta L_2$$) of rod ($$L_2$$) (since magnitude of $$L_2$$ is higher, $$\Delta L_2$$ would be higher) but my book says more force must be applied to rod with larger length ($$L_2$$) if we want to keep the ratios $${\Delta L_1\over L_1}$$ and $${\Delta L_2\over L_2}$$ same for both rods. So I think my concept isn't clear and I am wrong somewhere. Please correct me.

• What exactly does the book (which book?) say? – Claudio Saspinski Sep 13 '20 at 16:24
• @Claudio Saspinski I added a pic. Please help me. Book is Fundamentals of Physics by Halliday/Resnick /Walker – Forex007 Sep 13 '20 at 16:42

## 1 Answer

You are correct that—all else being equal—a longer rod will contract proportionally more under an axial load. In this case, however, it sounds like a greater-than-proportional contraction is required, as all rods are required to contract to the same final length. In other words,

$$L_4-\Delta L_4=L_3-\Delta L_3$$

$$\frac{\Delta L_4}{L_4}> \frac{\Delta L_3}{L_3}$$

To avoid confusion, it would be better if the book said something like "The single longer leg must be compressed by a larger relative amount (or larger strain) $$\Delta L_4/L_4$$ and thus by a force with a larger magnitude $$F_4$$."

Does this resolve the issue?