Chapter 3.7, Problem 100E

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y = 0 when the mass hangs at rest. Suppose you push the mass to a position y₀ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is

     $y=y_0\cos \left(t\sqrt{\frac{k}{m}}\right),$

where k > 0 is a constant measuring the stiffness of the spring (the larger the value of k, the stiffer the spring) and y is positive in the upward direction.

100.    Use equation (4) to answer the following questions.

  a.    The period T is the time required by the mass to complete one oscillation. Show that $T=2\pi \sqrt{\frac{m}{k}.}$

  b.    Assume k is constant and calculate $\frac{dT}{dm}$

  c.    Give a physical explanation of why $\frac{dT}{dm}$ is positive.