A mass with mass 4 is attached to a spring with spring constant 24 and a dashpot giving a damping 20. The mass is set in motion with initial position 4 and initial velocity 2. (All values are given in consistent units.) Find the position function

Explanation: In a mass-spring-damper system, the differential equation that rules the motion of the mass is: mx" + cx' + kx = 0

Using m = 4, k = 24 and c = 20, we have

4x" + 20x' + 24x = 0

Simplifying, we have

x" + 5x'+ 6x = 0

The characteristic equation of this differential is

The solutions for the quadratic equation are: and

Hence:

x(t) =

x'(t) =

To determine the constants, we have the initial conditions x(0) = 4 and

x'(0) = 2, then:

Substituing the constants:

The position function for this system is: