A mass weighing 4 pounds stretches a spring 4 inches.(a) Formulate an initial-value proble...

A mass weighing 4 pounds stretches a spring 4 inches.

(a) Formulate an initial-value problem that corresponds to the motion of this un-damped mass-spring system if the mass is extended 1 foot from its rest position and released (with no initial velocity).

(b) Using the result of Exercise, find the solution of this initial-value problem.

Exercise

Consider the equation

for the motion of a simple harmonic oscillator.

(a) Consider the function y(t) = cos βt . Under what conditions on β is y(t) a solution?

(b) What initial condition (t = 0) in the yv-plane corresponds to this solution?

(c) In terms of k and m, what is the period of this solution?

(d) Sketch the solution curve (in the yv-plane) associated to this solution. [Hint: Consider the quantity y2 + (v/β)2.]