A toothbrush is an interesting example of a force- limiting system that employs buckling in its basic design. When you brush your teeth, your hand applies a loading that essentially sets up a reaction force on the brush-head from the teeth which is distributed (approximately) in an even manner over the bristles of the brush as an axial load. Idealize the brush-head as a regular array of N bristles spread evenly over an area w by l, and determine an expression for the maximum force one can apply to the brush-head. Assume each individual bristle has a length h, radius r, and modulus E.