Show that the local maximum or minimum for the displacement

Show that the local maximum or minimum for the displacement of an underdamped oscillation does not

occur halfway between the times at which the mass passes through its equilibrium position. However,

show that the time period between successive local maxima (or minima) is constant. What is this

period?

Stuck with this problem please help

Solution

We are now in a position to formulate a model of a spring/mass system. By Newton’s Second Law, F = ma and we realize that a = y 00(t). So we obtain the second order differential equation my00 = ky which we rewrite as my00 + ky = 0 where m > 0 and k > 0. Of course we can solve this system for all values of m,k since it is a homogeneous linear second order DE with constant coefficients. It has general solution: y(t) = c1 cos(r k m t) + c2 sin(r k m t)