Nail tips exert tremendous pressures when they are hit by hammers because they exert a large force over a small area. What force must be exerted on a nail with a circular tip of 1.00 mm diameter to create a pressure of $3.00\times 10^9{\text{ N/m}}^2$ (This high pressure is possible because the hammer striking the nail is brought to rest in such a short distance.)

Video Transcript

This is College Physics Answers with Shaun Dychko. We want to know what force needs to be exerted on a nail by a hammer in order to exert a pressure of three times ten to the nine newtons per squared meter, given that the tip of the nail has a diameter of one millimeter, which I converted to meters by multiplying by ten to the minus three. So pressure is force divided by area, we can solve this for F by multiplying both sides by A, so force is the pressure times the area. So that’s pressure times pi r squared because this is the area of a circle, the circular tip of the nail, but we don’t know what the radius is but we can substitute it with diameter divided by two, which gets squared and the denominator two gets squared so we have four in the denominator. So we have pressure times pi times diameter squared over four, and this is going to be the force. So three times ten to the nine newtons per squared meter times pi times one to ten to the minus three meters diameter of the nail squared, divided by four, this gives 2360 newtons.