Question

An object of mass 3 grams is attached to a vertical spring with spring constant 27 grams/seca. Neglect any friction with the air. (a) Find the differential equation y = fly, y) satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y = (b) Find rı,r2, roots of the characteristic polynomial of the equation above. ru,r2 = (b) Find a set of real-valued fundamental solutions to the differential equation above. yı(t) = y(t) = (c) At t = 0 the object is pulled up 33 cm and the released with an initial velocity upwards of 9 cm/sec. Find the amplitude A> 0 and the phase shift OE(-1,T] of the subsequent movement. A=

Answer 1

• Griven: mass of the object= 3 grams Spring constant, k = 27 grams/5² • Solution: a) The general diffesential form, my'tßy'tky=fits => 3 y"+ Oy'+ 27y=0 = 3y" +27y=0 b) The .. y'=-9y auxiliary equation of diffesential equation, - y"+9y=0 m²+9=0 = m = 1 3; Ima 31,-3i) general solution of differential equation, y = C, Co53t + C2 singt - 0 y=6 y. S +6 Yz Cts :. (4,(t)= (0934 192CH= Sin 37 C) The

d) Given; y(0) = 303, y'CO) = 9 :y(0) = 363 Fromean o 353= 4 coso +C, sino 353=, *1+ 0 ..14=303) Now, y'CO) - 9 y'(t)= -34, sin3t +342.60934 :.y.co)= -34, sino +372 (OSO 9= 0 + 362 1. ( - q - 3 Hence, y= 313 605 3t + 3 sinat = veza9 (358, 60, ** payson 38) 6 ( v cos 3+ + ļ sin st) - 6 cos (3t-II) cos (a-b)= cosa . tsina sinb 84 po hng & ". Cosme Ve Sinteza y = Acos (wt+ )