Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

Poiseuille's Equation

Challenge Level:

Water is pumped at a steady rate through a straight circular pipe of length $L$ and radius $R$.

I make the assumption that if the flow is steady then the flow rate $Q$ of water through the pipe can only depend on $L$, $R$, the constant difference in pressure $\Delta P$  between the two ends of the pipe and the viscosity $\mu$ of the fluid. How strongly do you agree with the validity of this assumption from a practical point of view? Under what circumstances are they most likely to be valid?

The standard equation governing flow along a circular pipe is called the Poiseille-equation:

$$Q = k\frac{R^4 \Delta P}{\mu L}\, \mbox{for a constant } k$$
Show that this makes sense from the point of view of the physical units of the various variables. Would any other combinations of variables combine to give a flow rate?

How might you devise an experiment to determine the numerical value of $k$?

Suppose that you push water along a horizontal pipe into the air. If you have a fixed amount of force at your disposal, are narrow pipes or wide pipes best to get the largest flow rate?