Question

3a. [10pts) A 32 lb weight is attached to a spring whose constant is 25 lb/ft. Initially the mass is released I ft below the equilibrium position with a downward velocity of 1 ft/sec. Find the equation of motion 3b. 10pts) Determine the equation of motion in part(a) if the surrounding medium offers a damping force numerically equal to 10 times the instantaneous velocity. 3c. [14pts) Determine the equation of motion in parts(a)-(b) if the weight is driven by an external force equal to f(t) = 50 cos 10t

Answer 1

Given wh=32 lb k= 25 ebfft m= 32 slug El slug Equation of system malt ka=0 22" +25x=0 Auullary equation 1r2+25 = 0 r=&se X(t) = C, Cossa the sinsa Initial &'(0) = -1 condrtions n (0)=-1 x'(t) = -56, sinsx +562 sim coisa

BULLULTURE SUIVELHOORNURULUTU -1 -0+562 (2= -45 - X(t) = - COMIX - 5 sinsk (b) Given c=10 Equation of system ma" + cx'+kx =0 x"+lox' + 25% =0 Auxiliary equation rrior +25=0 (5+5)=0 8=-51-5 a(t) = ciest & catést Initial conditions a(0) = -1, r =-1 -15C,+o AGE-1 x'(t) = -5ce the taste 1 = -56, +(2 eq=r6 -5test. -'. x(+) = -e - 65 l t = 85€ -Gtest

(0) f(t) = 50 COS Clot) Equation of system ma" tcx'tkx = f(t) x"+lox' +251 = 50 cos Clot) Homogenow equation X" Hlox' +25% =0 Aurillady equation y? Hlor+25 =0 (r+5)=0 8=-5,- de acest totest xp = [ (50) Cos (tot) =-10 HOD+25 2 . so colliot) TOD.75 3 (0-100) cos(tot) : -? 11+ ons cooliot) Dato xp = (los (tot) - sin cot) ] .. x(t) = xe tap xce)= evettetést test (cos(601) – sin cit]

-39 . 33 a (0) --1 xloer) 15 city = a'z-sestrzest strett - 20 Sincrot) 7.0 (0) (0+) HE -54, +62 - 80 (2=-6 32 . X(t)= -sg 27 -5t e 5t 6te + ? 37 cos llot)-8 Sincrot) lol
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