Question

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

Answer 1

Answer:
`x(t) = 14e^(-2t) - 10e^(-3t)`

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

`r^(2) + 5r + 6 = 0`

The solutions for the quadratic equation are:
`r_(1) = -2` and
`r_(2) = -3`

Hence:

x(t) =
`C_(1)e^(-2t) + C_(2)e^(-3t)`

x'(t) =
`-2C_(1)E^(-2t) - 3C_(2)e^(-3t)`

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

x'(0) = 2, then:

`x(0) = C_(1) + C_(2) = 4\\ C_(1) = 4 - C_(2)`

`x'(0) = -2C_(1) -3C_(2) = 2\\-2(4-C_(2)) -3C_(2) = 2\\C_(2) = -10\\C_(1) = 4 - C_(2)\\C_(1) = 14`

Substituing the constants:

`x(t) = 14e^(-2t) - 10e^(-3t)`

The position function for this system is:
`x(t) = 14e^(-2t) - 10e^(-3t)`