1-kilogram mass Is attached to a spring whose constant Is 21 N/m, and the entire
ID: 2866388 • Letter: 1
Question
1-kilogram mass Is attached to a spring whose constant Is 21 N/m, and the entire system Is then submerged in a liquid that imparts a camping force numerically equal to I times the instantaneous velocity. Determine the equations of motion if the following is true. (a) the mass is Initially released from rest from a point 1 meter below the equilibrium position x(t) = [ ] X m (b) the mass Is initially released from a point 1 meter below the equilibrium position with an upward velocity of 1 1 m/s x(t) = [ ] mExplanation / Answer
If
ma + cv + kx = 0
The discriminant is
D = c^2 - 4mk
Here,
c = 10, m = 1, k = 21. Thus,
D = 16 > 0
Thus, as D > 0, it is overdamped, and has an equation
x(t) = A e^(Bt) + Ee^(Ft)
where, A, B, E, F are constants to be determined.
where
B = [-c + sqrt(D)]/2m = -3
F = [-c - sqrt(D)]/2m = -7
The initial conditions are
x(0) = -1
x'(0) = 0
Thus, A and E must satisfy
A + E = -1
-3A - 7E = 0
Thus,
A = -1.75
E = 0.75
Thus,
x(t) = A e^(Bt) + Ee^(Ft)
becomes
x(t) = -1.75e^(-3t) + 0.75e^(-7t) [ANSWER, PART A]
*******************************
If, instead, it has x'(0) = 11, then our equations for A and B will instead be
A + E = -1
-3A - 7E = 11
Thus,
A = 1
E = - 2
Thus,
x(t) = e^(-3t) - 2e^(-7t) [ANSWER, PART B]
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