Chapter 3.7, Problem 99E

 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

  $y=y_0\cos \left(t\sqrt{\frac{k}{m}}\right)$,    (4)

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

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

a.    Find the second derivative $\frac{d^{2}y}{d{t}^2}$.

b.    Verify that $\frac{d^{2}y}{d{t}^2}=-\frac{k}{m}y$.