Answer to Question #174283 in Differential Equations for MOHAMMED

Question

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.

Answer 1

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

$y'=-3/2e^{-3/2t}costC_1-e^{-3/2t}sintC_1-3/2e^{-3/2t}sintC_2+e^{-3/2t}costC_2=e^{-3/2t}(-3/2 costC_1-sintC_1-3/2sintC_2+costC_2)$

$2=-3/2C_1+C_2$

$C_1=C_2=-4$

$y=-4e^{-3/2t}(cost+sint)$