a 16lb object stretches a spring 89 feet by itselfthere is n

a 16lb object stretches a spring 8/9 feet by itself.there is no damping and no external force acting on the system.the spring is initially displaced 6 inches upwards from its equilibrium position and given an intial velocity of 1 feet/second downward.find the displacement at any time,t

Solution

mass = weight/acceleration = 16/32 = 0.5

k = mg/L = 16/(8/9) = 18

The equation will become

1/2u\'\' + 18u = 0

u\'\' + 36u = 0

Initial conditions u(0) = -6/12 = -0.5, u\'(0) = 1

Writing the characteristic equation of homogeneous differential equation

p^2 + 36 = 0

p = -6i.6i

u(t) = c1cos(6t) + c2sin(6t)

u(0) = -1/2 = c1

u\'(0) = 6c2 = 1

c2 = 1/6

Hence the final solution to the differential equation will be

u(t) = -1/2*cos(6t) + 1/6*sin(6t)