Air spring. The elasticity and bulk modulus of air can be measured using an apparatus consisting of a vertical glass tube and a ball bearing that just fits in the inner diameter of the tube. A 110-g ball bearing is dropped into the top of a 47-cm-high glass tube of diameter 3.0 cm. The ball is observed to oscillate as if the air column were a spring, 59 times in 10 s giving a period of 0.17 s. 

(a) Find the effective spring constant of the air column.

(b) Show that this effective spring constant is related to the bulk modulus by k = A²B/V, where A is the cross-sectional area of the tube and V is the volume of the tube. Start with Equation 11.25 and express the change in volume in terms of the displacement of the mass. 

(c) Determine the bulk modulus of air.