Balance and inertia

How do gymnasts keep their balance on a beam? Why do tightrope performers carry a long pole? Try standing with your feet placed one behind the other.

Why is this more difficult than standing normally, do you think?

When balancing, your stability is increased by:

• being symmetrical about the centre
• distributing mass far away from the centre
• lowering the centre of gravity
• widening the base of support

The mathematical concept involved here is that of inertia. An object which has high inertia is slow to move, an object which has low inertia is easy to move. Mathematically, inertia is related to the mass of the object, its average radius, and the way that mass is concentrated relative to the centre.

In this equation, the constant C shows how much the mass is concentrated towards the centre of the object. A high value indicates that mass is concentrated towards the outside of the object, a low value that it is concentrated towards the centre of the object, as can be seen in the diagram on the left. The value of C for the hollow shell is 2/3, whereas for a solid sphere of the same mass (different density), it is only 2/5.

Sports where inertia is a factor to consider are skating, diving, cycling, running and tennis. When skaters want to move more rapidly, they put their arms close to their bodies, and when they want to move more slowly, they stretch their arms out.

Ordinary cycle wheels look like those in the left-hand diagram while racing cycle wheels look like the right-hand diagram.

In an ordinary cycle wheel, the mass is concentrated towards the outside, as with the hollow shell. This increases the value of C, and thus the inertia, helping us to maintain our balance. The spokes mean that there is turbulence in the air around the wheel, however, and racing cyclists therefore prefer solid wheels, like that on the right. The inertia is also less, helping them to move more quickly.

The other sports mentioned, running and tennis, also involve inertia. Runners do not try to run with their arms dangling down by their sides, but tuck them up into their bodies. This reduces their inertia, so that they can move more quickly. Modern tennis rackets have a lot of their weight at the end of the racket, so increasing their inertia, compared to the old-fashioned wooden ones. The purpose of this is to prevent the player's wrist being affected so much when striking the ball.

Next: High Jump