A 16 lb weight is attached to a spring whereupon the spring

A 16 lb weight is attached to a spring whereupon the spring is stretched 1.6 ft and allowed to come to rest. The weight is set into motion from rest by displacing the spring 9 in above its equilibrium position and also applying an external force F(t) = 5 cos(2t). Find the subsequent motion of the weight if the surrounding medium offers a resistance of-2x lb. (The resistance is X^(.), but chegg does not allow me to enter this equation.)

Solution

ok so first some equations
F=ma; F=ks; my\'\'+By\'+ky=F(t); m=mass; k=spring constant; B = resistance

alright so your mass is 16lb = (m)(32ft/s^2) m=1/2slugs

and your spring constant = k=F/s ====> 16/1.6 k=10

and I am guessing your B is equal to 2? I am not sure what you mean with the resistance.

alright so now just sub all your stuff into the equation

1/2y\'\' + 2y\' + 10y = 5cos(2t)

put in standard form:

y\'\' + 4y\' +20y = 0

quad rule

(-4 ± 16-80)/2    ====> -2 ± 4i   =====> e^-2t(c₁cos4t + c₂sin4t) = y_c

ok now you need to solve for y_p

I would recommend super position in this case if you know it, if not just do variation of parameters

I did not solve all the way through as I was unsure if you reall meant 2 as your resistance or I just didn\'t understand what you meant. But if you have something else as your resistance just follow the above and it will be pretty easy.

Also your intial conditions for this problem should be

y(0) -9/12ft

y\'(0) = 0

I hope this helps some.