### Building up Friction

A series of activities to build up intuition on the mathematics of friction.

### Uniformly Unstable

Invent shapes with different numbers of stable and unstable equilibrium points

### Why do this problem?

This problem will develop students' intuition and skill with vectors and force laws in a context which encourages exploration and does not require any calculation to get started. By trying to construct stable configurations students will be led towards the notion of static points only occurring at places of zero electrostatic potential.

### Possible approach

This problem is very open and well suited to discussion to get things started. The main requirement is the encouragement that students start to try out specific configurations and then use the force law to determine whether or not the configuration will remain static.

More general statements are, of course, more subtle and difficult to analyse than particular examples. Clear, systematic thinking will be needed, as will a clear representation system (diagrammatically or algebraically).

After doing this problem, you might refer students to 'Earnshaw's theorem' http://en.wikipedia.org/wiki/Earnshaw%27s_theorem

### Key questions

Are there any obvious simple cases to look at?

Can you make any symmetrical configurations to look at?

What is special, if anything, about configurations in which $1$ or more particles don't move?

How might you attempt to fix down a configuration which will move by adding more fixed particles?

### Possible extension

Take an entirely stable configuration. Consider the question: what happens if you nudge one of the particles slightly?

### Possible support

Concentrate on intuitive attempts at the problem.