Question

In: Physics

A mass m=4 is attached to both a spring with spring constant k=145 and a dash-pot...

A mass m=4 is attached to both a spring with spring constant k=145 and a dash-pot with damping constant c=4.

The ball is started in motion with initial position x0=1 and initial velocity v0=6 .
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 x(t)=c1e^(-ρt)cos(ω1t-α1) . Determine c1, ω1, ρ and α1.

c1=?
ω1=?

ρ=?

α1=?

Graph the function x(t) together with the "amplitude envelope" curves x=-c1e^(-ρt) and x=c1e^(-ρt).

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 u(t)=c0cos(ω0t-α0). Determine c0, ω0,and α0.

c0=?
ω0=?

α0=?

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