**PURPOSE: **To demonstrate how pulling angle affects the frictional
force, and to show that the minimum force required to pull an object
occurs when pulling at the angle of repose *a*, where the
coefficient of friction *u*=tan*a*.

**DESCRIPTION: **A wooden board of weight w, lying on a sandpaper
surface, is pulled at an angle *a* by a string connected to a spring
scale. The force F required to move the board is given by: F = w / [
(1/*u*)cos*a* + sin*a*]. Differentiating F with respect
to *a*, the minimum force is seen to occur at the angle of repose,
*u*=tan*a*. Pulling horizontally is definitely not the most
efficient angle!

**SUGGESTIONS: **Pull horizontally to determine *u*, then
check the angle of repose by tilting the sandpaper surface. Then pull
at a variety of angles to demonstrate that pulling at the angle of
repose requires the least force.

**REFERENCES: **(PIRA unknown.)

**EQUIPMENT: **Board with spring scale and string, sandpaper
base wih clamp to attach to lecture table, demonstration protractor.

**SETUP TIME: **5 minutes.