What does this equation mean?

Question

$$\propto \dfrac {\text {Cross-Sectional Area}}{\text {Volume}} $$

I've seen this equation on my textbook (physics).

Let us imagine that there are two pen with $7$ and $3$ lenght. Which is harder to break? According to the equation, $3$ is harder to break. Can you explain it?

Kindest Regards!

Answer 1

I suppose your statement is correct.

If $$\text{Cross-Sectional Area}\times \text{Length} = \text{Volume}$$ then, $$x\propto \frac{\text{Cross-Sectional Area}}{\text{Cross-Sectional Area}\times \text{Length} }$$ $$\therefore x\propto \text{Length}^{-1}$$

So, if $x$ is a measure of how hard something is to break, then, yes, something of three units length will be comparably harder to break than something of seven units length.

Answer 2

The $\propto$ symbol represents proportionality. So, whatever is on the left hand side is directly proportional to cross-sectional area and inversely proportional to volume.

Now, as to the actual relationship you are asking about, that's impossible to confidently answer given how broad your question is. That being said, that relationship is correct for two of the most common failure modes: bending and compression (the third being tension, which isn't a function of element length).

For elements under transverse loads, bending moments are generated which are a function of the element's length (and of many other things). So longer elements suffer greater bending moments under the same load, and can therefore withstand lower loads than (longitudinally) shorter elements. This can be easily displayed with a toothpick: get one and snap it in half; get one of the halfs and repeat; repeat to satisfaction; you'll notice it gets harder the shorter the segment.

Elements under compressive loads frequently collapse not to due excessive compressive stress, but due to buckling. An element's buckling load is inversely proportional to its length, so (longitudinally) shorter elements under compression can withstand larger loads than longer ones.