Question

here is the question, please help me with this question

When a 5 kg mass is attached to a spring whose constant is 180 N/m, it comes to rest in the equilibrium position. Starting at t= 0, a force equal to f(t) = 20e cos 3t is applied to the system. In the absence of damping, -5t (a) find the position of the mass when t=1. (b) what is the amplitude of vibrations after a very long time?

Answer 1

-5t fit) = 20e When a 5 kg mass is attached to a spring whose constant is 180 N/m, it comes to rest in the equilibrium position. Starting at t=0, a force equal to cost is applied to the system. In the absence op damping. s-spring differential equation (with no demping) is myutky = f(t) flt/= 20e Lost The mass st given, m=5, K= 180 st 54"+180 y=20e coszt dividing by s .-5t Y"+36y=46 cogut • To find the homogeneous solution y(t): The auxilliary equation is 8²+36=0 → 8²=-36 8=+6i .: I (t) = 4cos et +ęsiast • To find the particular solution y(t): I 1+2= "cost where Dad D²+36 de = 4. Ietcost D²+36 4e-st -5t =4.e-st eat alt! Cost (0-5)+36 F (D) at I re OLD FID+*)

st =4e cost D²10D +61 - -46 est . il cosat cost -3²10D+6l FLD²) 1 cosat 1 = 46-5t Fl-a3 cosat S2-10D =40 St (52+10D) cost 52²-loop? = 46 st. 1524100) .cos3t 522100(-32) = 46 st (52H0D) cosst 3604 - 1 e-st (5200szt-30 sinat) go! e 5t(s2w0szt-20fiu3t) . the general solution is YLt) = 4, cost-tqsinstt. pol given, y(0)=0 :: 0=G.COJCol+ç bico) + societº (52090-30440) o=C to +52 →=-52 901 90) • Y' (t) = -66, Sinbt +6C coset+ Lest St (-isc Sin3t-90983+) - E est (522653t-30 siust 901 901

• given, y'lo)=0 0= –66, sinot 6gasot Leo (-156 Sino-go coso) { e(52650-30 sino) 901 5 90) ► 02 0+642-90 246 gol gol 69=2350 =175 gol 2703 putting these values of G andų in O, we get y(t) = -52 cosst! 175 sinstat e St (526033t-30 finzt) gol 2703 901 • when tz y (2) = -52 costa- 175 sabatt é 901 2703 (5260332-30 sin3t) you = -52 Sol .(1- 175 (0)+ -SA e 901 2703 ? (52(-1)-30(0) 52 (lte 52) - - 52 52 é sa - gol 901 gol -st (6) As ta, e .. y(t) - 52 yol cosat -175 2703 singt Amplitude = {-27 + (-1757 Jol 35901