Question

A 2 kg mass is attached to a spring having spring constant 10 N/m. The mass is placed in a surrounding medium with damping force numerically equal to 8 times the instantaneous velocity. The mass is initially released from rest at 1 meter above the equilibrium position. Find the equation of motion. 

Answer 1

From the given data we can construct the differential equation as

$2x'' + 8x' + 10x = 0$

with initial conditions $x(0) = 1$

$x'(0) = 0$

Auxiliary equation can be

$2m^2 + 8m + 10 = 0$

On solving we get,

$m = -2+i, -2-i$

Hence C.F = $e^{-2t}(C_1cost + C_2sint)$

After applying initial conditions we get

$C_1 = 1$

and, $C_2 = 2$

Hence the solution of the differential equation is

$x(t) = e^{-2t}(cost + 2sint)$