# Consider the following model steel bridge. What would the real bridge weigh and would the real bridge sag?

## Consider a model steel bridge that is 1/100 the exact scale of the real bridge that is to be built. If the model bridge weighs 50 N, what will the real bridge weigh? If the model bridge does not appear to sag under its own weight, is this evidence the real bridge, built exactly to scale, will not appear to sag either?

Mar 16, 2017

Because of the 'square-cube rule', the bridge will be 10,000 times as strong as the model but weigh 1 million times more. The fact that the model does not appear to sag does not guarantee that the bridge will not appear to sag.

#### Explanation:

This is an example of the 'square-cube rule': as things change scale, cross-sectional area (and surface area) changes as the square of the length, while volume (and therefore mass (if the density of the materials is the same) changes as the cube of the length.

So if the real bridge is 100 times as large as the model, its mass will be ${100}^{3}$ = $1 , 000 , 000$ times the mass, = $50 , 000 , 000$ or $5.0 \setminus \times {10}^{7}$ $N$.

The strength of the beams holding up the bridge will be related to their cross-sectional thickness, which will increase as the square of the length, so they will be ${100}^{2}$ = $10 , 000$ times as strong.

Because the bridge is 1 million times as heavy but only 10,000 times as strong as the model, the fact that the model does not appear to sag is no guarantee that the bridge will not appear to sag.