A mass with mass 4 is attached to a spring with spring constant 25.5625 and a da

Question

A mass with mass 4 is attached to a spring with spring constant 25.5625 and a dashpot giving a damping 20. The mass is set in motion with initial position 6 and initial velocity 0 All values are given in consistent units. Find the position function z z(t) The motion is (select the correct description) A. overdamped B. critically damped C. underdamped O If the system is underdamped, rewrite your answer in the form x(t) = 6 If your answer is not underdamped, enter "N" in each of these answer blanks. cos(

Explanation / Answer

We have given mass m=4,spring constant k=25.5625 dumping constant c=20 ,initial position x(0)=6 and initial velocity v(0)=0

the discriminant of the characteristic polynomial is c^2-4mk=(20)^2-(4*4*25.5625)=400-(16*25.5625)=-9<0

so the motion will be underdamped

the roots of the characteristic polynomial equation 4r^2+20r+25.5625=0

r=[-20+/-sqrt(400-4*4*25.5625)]/8

r=[-20+sqrt(400-(4*4*25.5625))]/8,r=[-20-sqrt(400-4*4*25.5625)]/8

r=[-20+sqrt(400-409)]/8,r=[-20-sqrt(400-409)]/8

r=[-20+sqrt(-9)]/8,r=[-20-sqrt(-9)]/8

r=(-20+3i)/8,r=(-20-3i)/8

so the solution is x(t)=e^((-20/8)t)[c1cos(3t/8)+c2sin(3t/8)]

pligging the initial conditions x(0)=6

c1e^(0)cos(0)+c2e^(0)sin(0)=6 implies c1=6

x'(t)=e^((-20/8)t)[-(3/8)*c1sin(3t/8)+(3/8)*c2cos(3t/8)]+[c1cos(3t/8)+c2sin(3t/8)]*(-20/8)*e^((-20/8)t)

pluging v(0)=0

e^(0)[-(3/8)*c1sin(0)+(3/8)*c2cos(0)]+[c1cos(0)+0]*(-20/8)*e^(0)=0

plug c1=6 into above equation

6[0-(20/8)]+c2[(3/8)-0]=0

-120/8+(3/8)c2=0

(-120+3c2)=0

3c2=120 implies c2=40

plug c1=6,c2=40 into x(t)

x(t)=6e^((-20/8)t)cos(3t/8)+40e^((-20/8)t)sin(3t/8)

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