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

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Physics

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

The ball is started in motion with initial position x0=1and initial velocity v0=6 .
Determine the position function x(t) .

x(t)=?

Note that, in this problem, the motion of the spring isunderdamped, therefore the solution can be written in the formx(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 positionand velocity, but with the dashpot disconnected ( so c=0). Solvethe resulting differential equation to find the position functionu(t).
In this case the position function u(t) can be written asu(t)=c0cos(ω0t-α0). Determinec0, ω0,and α0.

c0=?
ω0=?

α0=?

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