Question

Consider a spring of mass 1 Kg attached to a spring obeying Hooke's Law with spring constant K

Problem 4. (15 pts) Consider a spring of mass 1 kg attached to a spring obeying Hooke's Law with spring constant k N/m. Suppose an external force F(t) = 2 cos 3t is applied to the mass, and suppose the spring experiences no damping. Suppose the spring can be displaced 0.2 m by a 1.8 N force. If the spring is stretched 1 m from equilibrium initially and is then released (at time t=0), set up and solve an IVP for the displacement X(t) of the spring from its equilibrium position. 1

Answer 1

from Newton law {F=mc²x dt2 EF= Fspring + Fexternal -☺ ma²x - kx + Fex d42 {no damping forcel m d²x + Kx = Fex dt + m dax (K 10 = Fex com c -0 Idt2 im K= spring constant, me mass ya F = KX (1.8)= (k) (0.2) k= 1.8 9 N/m As 0.2 m= 1kg Putting values in o d²x +92 = 2 Cos 3 t dt2 - . clax dt2 + 9x = = 2 cos3t x"+9x=20053t/ initial value problem. Auxilany equation Ded (comb lignentally function (0%+9)x = d cos 3+ m +9=0 ME-9 ct ² m=13

Solution of this CF X(t)= A cos[3 +] + 8 Sin (3 +) A, 8= constant Initial condition Im At t=0, spoing stretched X(b)= im by Put It=0 in (2) X 10) = 1 = A[J+ 8L0] A=1 as it is released from rest '(o)=6 diff- (2) Worot t dxt) = -33 Asin (3+) + 38 cos3+) dt x'(t) = - 3 Asin 13t) +3 BLOS (3+) At HEO], x't)= x'(0)=0 0= - 3A Sin (320)] + 38 C08 (360)] B=0 (I= C063t 80 C.F.is Finding panti cular solution. coscat) I coslat) P.I= #10) fta²) if fl-a?)=0 I

than (A) P-I = ti Cosa t) ft-a²) I 2008 3 t) Il 079 Id Cogl3+) [E-37797 (becomes zero) using @ P-I= 2 Cos 3 t) at L cos3t 20 P. I= t scos3t = = t Sin3 t) 3 Total = CF + P.I solution to X(t)- Cos(3 t) +ts in (3 t) 3 De forf Oberos de Anl