A mass m = 4 is attached to both a spring with spring constant k = 485 and a dash-pot with damping constant c = 4.

The ball is started in motion with initial position and
initial velocity .

Determine the position function x(t).

x(t)=

Note that, in this problem, the motion of the spring is underdamped, therefore
the solution can be written in the form . Determine , ,and .

Graph the function x(t) together with the "amplitude envelope" curves
and .

Now assume the mass is set in motion with the same initial position and
velocity, but with the dashpot disconnected ( so c = 0).
Solve the resulting differential equation to find the position function u(t).
In this case the position function u(t) can be written as
. Determine
, and .

Finally, graph both function x(t) and u(t) in the same window
to illustrate the effect of damping.

This problem is an example of critically damped harmonic motion.
A mass m =8 is attached to both a spring with spring constant k =392 and a dash-pot with damping constant c = 112.

The ball is started in motion with initial position and
initial velocity .

Determine the position function x(t).

x(t)=

Now assume the mass is set in motion with the same initial position and
velocity, but with the dashpot disconnected ( so c = 0).
Solve the resulting differential equation to find the position function u(t).
In this case the position function u(t) can be written as
. Determine
, and .

Finally, graph both function x(t) and u(t) in the same window
to illustrate the effect of damping.

This problem is an example of over-damped harmonic motion.
A mass m = 4 is attached to both a spring with spring constant k = 112 and a dash-pot with damping constant c = 44.

The ball is started in motion with initial position and
initial velocity .

Determine the position function x(t).

x(t)=

Graph the function x(t).

Now assume the mass is set in motion with the same initial position and
velocity, but with the dashpot disconnected ( so c = 0).
Solve the resulting differential equation to find the position function u(t).
In this case the position function u(t) can be written as . Determine , and .

Finally, graph both function x(t) and u(t) in the same window
to illustrate the effect of damping.