# Why does the change in length depend on the initial length of a rod?

When a rod of some length is heated then it undergoes linear expansion (also areal and volumetric) but I don't understand Why does the amount of expansion depend on the initial length of the rod ?

If we have two rods one of half the length of the other, both of same material and at same initial temperature then if both are heated to same final temperature then the one with greater length undergoes greater change in length. This is what confused me.

Shouldn't the energy per unit length be same in both the rods and so each bonds should get the same energy in both the rods and so both should expand the same amount?

• Each unit of length expands the same amount. If a 1 m rod expands by 1 cm, and you put 2 of them end to end, to make a 2m rod, the total expansion is 2 cm. Aug 20, 2021 at 5:20

Now, each bond has been given the same amount of energy and will expand by a tiny amount $$\Delta{}x$$. But the longer rod has more bonds, and each bond is expanding by a tiny amount. The smaller rod expands by $$\frac{L}{b}\Delta{}x$$ and the larger rod expands by $$\frac{2L}{b}\Delta{}x$$ where $$b$$ is the length for a single bond, so that $$L/b$$ is the total number of bonds along the rod's length.