Consider a body of mass attached to a spring with spring constant . The differential equation , where is the vertical displacement, can be used to model the dynamics of this system. However, such a differential equation implies that, once the body starts moving, it will oscillate forever. A better model is one that also considers the effects of frictional forces. If we add a frictional force that is proportional to the velocity of the body, we get the following differential equation where .
Letting and , there are three possible behaviours of the solution to this equation which we can define in terms of and as follows: over damped when ; critically damped when ; and underdamped or oscillatory when . Which of the following solutions can be used to describe one of these behaviours:
- , where and
The one-dimensional time-independent Schrodinger equation is given as
where is a wave function which describes the displacement of a single particle of mass , is the total energy, is the potential energy, and is a known constant. Since for the particle-in-a-box model, where , this second-order differential equation becomes
Find the general solution for this differential equation.
Consider a mass that oscillates at the end of a spring having a spring constant . The following second-order differential equation describes the vertical displacement of this spring-mass system: This differential equation neglects the influence of air resistance or frictional forces. Find the general solution which describes the vertical displacement of this spring as a function of time .