Question

1) A force of 2 N stretches a spring 0.5 meters. The mass of 1 kg is attached to the spring and set into motion in a medium that offer a damped force equal 4 times the velocity. If the mass is at stating from 0.5 m above the equilibrium position with a downward initial velocity of 0.2m/sec a) Find the equation for the position if the system is exerted by an external force of f(t) = 4 cost. b) Estimate the position of the mass at t=1t/8 and t = 70/4 sec Describe the motion after a long period of time c)

Answer 1

wa Spring Constant 2 / 3 = 4 N/m if 4/o)= -.5 alt the instantaneous displacement the spring - system from equillitrium, the differential equation of motion is d? da dd2 9.8 4 at 42+2 Cost + 4 +4x = 9.8 d22 + Cost characteristic equation in 8 + 40 + 400 7 (8+2)² 26 7 8 = -2 -2 Complementary function is a & 2 (G+ St) integral (9.8+4 Cost) 8²440+4 2.7 + 4 Cost 40 +3 D- Cojla Particular 7 1 2.7 + 0²_2 1 2.7 + - x = é 27(6+61) + 2.77 (0) -5 G +2.7 t x'[1] = {2+ (- 24-261) + Cut - fin 2(E) - o 21/223 2.7 4 (- Sint 3 Cost) 13 4 4 Sint 3 Cost + 13 13 4 3 t 13 13 3 223 - - 15 7 G = 13 65 Cost 3 sint 13 13 4 2 13 7 G - 2+29 13 157 1-446 t) + 2.7 5 15 13 453 - + + 4 lint 3 asb 151 15 13 223 65 1515 + 120) 223 ..99 (1/8) - е ) + 21 7 + 4 3 in (h to 63(978) E.34 x(5/4) +227 + 1/23 ford, alt) – 2.7 + 73 Cost sint. There ofter is simples harmonic + 1570 30 13 motion is long time