Hooke's law and String Vibration:

1. Hooke's law. A massm is hanging from a spring of natural lengthx0 and spring constant k. Due to the effect of gravity, the equilibrium length will not be x0, but the spring is stretched some additional lengthd until it reaches the new equilibrium length x0=x0+d.

a) Findd in terms of m, k, and g.

b) Show that the net force acting on the hanging mass can be written in the form F=k(xx0). (This result is somewhat curious; it tells us that, despite the presence of gravity, the net force is still that of a spring, just with a different "natural length.")

2. String vibration. A guitar has strings of length L=63.5cm. The material of the strings is nylon (nylon=1.15cm3g). The first string has a diameter d=0.254mm. The standard tuning for the first string is the note E4, whose frequency is 329.63Hz. Find the tension required to achieve this tuning, assuming that the guitar strings vibrate with a single antinode.