A 6lb weight is attached to a spring suspended from a ceiling. The weight stretches the spring 4 inches. The external force is f(t)=27sin(4t)-3cos(4t). The medium offers a resistance of 3dy/dt (ft/sec). Find the equation of the motion.

displacement x = - 0.046sin4t +0.006cos4t

Step-by-step explanation:

The model of the equation of motion is a forced motion equation and to determine the displacement of the weight as a function of time; we have:

the weight balances of the  elastic force in the spring  to be expressed by the relation:

mg = kx

where;

x=4 in (i.e  1/3 ft )

mass m = 6lb

let make k the subject; then:

k = mg/x = 6×32/(1/3) = 576

assuming  x to be the displacement form equilibrium;

Then;

(since F(t)=27sin 4t-3cos4t somehow faces downwards, mg=downwards and k(x+1/3)= upwards and medium resistance 3v =  upwards)

SO;

Assuming :

replacing  these values in the above equation

equating sin and cos terms

a = - 0.046 ;  b = 0.006

displacement x = - 0.046sin4t +0.006cos4t

combine all like terms then its just a 2 step equation

cause it equals more