A 2 kg mass is attached to a spring having spring constant 8N/m. The mass is placed in a surrounding medium with damping force numerically equal to 6 times the instantaneous velocity. The mass is initially released from the equilibrium position with an upward velocity of 2 m/s. Find the equation of motion.
From the given data we can construct the differential equation as
2x'' + 6x' + 8x = 0
with initial conditions x'(0) =2
D=36-64=-28
"t_1=\\frac{-6-\\sqrt{-28}}{4}=\\frac{-3-\\sqrt{7}i}{2}"
"t_1=\\frac{-6+\\sqrt{-28}}{4}=\\frac{-3+\\sqrt{7}i}{2}"
The C.F is "y=e^{-3\/2t}(C_1cost+C_2sint)"
By initial conditions
"y'=-3\/2e^{-3\/2t}costC_1-e^{-3\/2t}sintC_1-3\/2e^{-3\/2t}sintC_2+e^{-3\/2t}costC_2=e^{-3\/2t}(-3\/2 costC_1-sintC_1-3\/2sintC_2+costC_2)"
"2=-3\/2C_1+C_2"
"C_1=C_2=-4"
"y=-4e^{-3\/2t}(cost+sint)"
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