Question

A mass weighting 32 lbs stretches a spring 8 inches. The mass is in a medium that exerts a viscous resistance of 24 lbs when the mass has a velocity of 2 ft/sec.

Suppose the object is displaced an additional 8 inches and released.

Find an equation for the object's displacement, u(t), in feet after t seconds.

Answer 1

# W = 32lbs., z=8 anch= 8 geht 32x12 lbs/ft = 48 lbs/ft 12 W = kx o 8 friscono = c 24 2 =12 Absos ft xlo= i (0)=0 .ft 12 ca & kx Imm E Fama -(Kx + ci)=mäe met kr+ci bx tomä atc at K x 20 =azin m W ил 32 à + 12 x 32 2 à 48 x 32 2 x 20 32 = x + 12.075 å +48.3 x=0 Let x=emt be soln

The complementary function m2 + 12.675m 48.3=0 me - 6.0375 +3 uur m2 = - 6.0375 -3.442 + Bemat mit . The soln = x(+) = 4e x= é - 603752 [ ci cos 3.4ut + disin uut] xlo)= using Bic 8 12 6.0375t [ ä(t) = &- 6:0375! x- 6:0375 [c, cas 3.442 +d, sin 3.446] -3.444 sin3.net+ 3.und cos 3.44€] vi (0) -6.0395 [ci] 3.44d, 6:03750 ds = 1917 + 3.44 Sin3 [ Aus - 6.0375t aix(+)= e Cos3. uut + 1.17 Sinsiuut 12 6.0375t [ & Cassiuut + 1017 Sins uut ft Am 12 a(t) = 6: Ult)= e