A mass weighing 4 lb stretches a spring 1.5 in. The mass is given a positive displacement of 2 in from its equilibrium position and released with no initial velocity. Assuming that there is no damping and that the mass is acted on by an external force of 2 cos 3t lb, formulate the initial value problem describing the motion of the mass. (A computer algebra system is recommended.)

u¹¹ + 256u = 16cos3t, u(0) = 1/6, u¹(0) = 0, u in ft, t in s.

Explanation:

given the values are:

ω = 4lb,

L = 1.5 in = 3/2.12 ft = 1/8 ft,

u(0) = 2 in = 2/12 ft = 1/6ft,

u¹(0) = 2 in = 2/12 ft = 1/6ft,

u(0) =  0ft/s,

γ -= 0 lbs/ft,

F(t) = 2cos3t.

From ω = mg we get

m = w/g = 4lb/32 ft/s² = 1/8lbs²/ft

and from ω = kL

k = ω/L = 4lb/ 1/8ft = 32lb/ft.

so therefore the initial value problem which describe the motion of the mass is given by

1/8 u¹¹ + 32u = 2cos3t, u(0) = 1/6, u¹(0) = 0.

equivalent to u¹¹ + 256u = 16cos3t, u(0) = 1/6, u¹(0) = 0