A mass with mass 7 is attached to a spring with spring constant 48 and a dashpot

Question

A mass with mass 7 is attached to a spring with spring constant 48 and a dashpot giving a damping 50. The mass is set in motion with initial position 9 velocity 3. (All values are given in consistent units.) Find the position function z(t) = The motion is (select the correct description) A. critically damped B. overdamped C. underdamped Finally find the undamped position function a(t) = G,cos(wt-ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c-0) u(t)

Explanation / Answer

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m = 1 , k = 36 , c = 12

So the differential equation is:

mx" + cx' + kx = x" + 12x + 36 = 0

with x(0) = 7 , x'(0) = -3

1)

The characteristic equation of the differential equation is:

r^2 + 12r + 36 = 0

-> (r+6)^2 = 0 -> roots are -6 , -6

So the position function is:

x(t) = Ae^(-6t) + Bt.e^(-6t)

x(0) = A = 7

x'(0) = -6A+B = -3 -> B = 39

So x(t) is:

x(t) = 7e^(-6t) - 39t.e^(-6t)

2)

zeta = c/2sqrt(mk) = 12/2*6 = 1

-> the motion is critically damped.

3)

c = 0 -> mu" + ku = 0 -> u" + 36u = 0

The characteristic equation is r^2 + 36 = 0

-> roots are -6i , 6i

So the undamped position function is:

u(t) = c0 . cos(6t - y0)

u(0) = c0.cos(-y0) = 7 -> c0.cos(y0) = 7

u'(0) = -6c0.sin(-y0) = -3 -> c0.sin(y0) = -0.5

-> tan(y0) = -0.5/7 -> y0 = arctan(-1/14) = -0.0713

c0 = 7/cos(0.0713) = 7.0178

So the position funcion is:

u(t) = 7.0178 cos(6t + 0.0713)

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