Question

Solve the damped harmonic motion system * + 2ki+w²x = 0 with initial conditions i = V at = 0 in the cases (i) wa = 10k2 (ii) wa = k2 (iii) k = 3,w2 = 5. Identify the type of damping, sketch the curve of u against t > 0 in each case, find zeros of x(t) when applicable.

Answer 1

JI S Damped liven Damped Oscillation is 4+2kätw²x = 0 - 0 and we know the differential equation & Oscillation is d' x + b dat n=0 - on comparing we get at case 1 :- w210k? from ea. O de + 24 d 7 + W x = 0 solution of this ego n = to è buttan sim (wit +0 1. where w'= [w2fble here we have w2= 10k2 t = 2K. Hence this type of oscillation is known as Underdumped damped Oscillation, to ) Shot on OnePlus By Aastha

Case II;w?=k2 again from eq. we have O 2 3 + 2 k dn + w?a=0 Solution of this eq. is. x = Ao e-kt sin (wit to) where w'a Suo? - (bm) and called where w?=k? when whalb 12 then this condition is critically damped Oscillation Case II R=3 Rw ²5 again from ea. O we have d2a + 2k dn + w²x = 0 d2n + 6 dn +50=0 at 2 at olan here b = 6, 4 = 5. solution of eain Ao e 3 in (w't + p). Shot on OnePlus By Aastha

where w' = JW2-(ben)? here 12 Hence in this condition Oscillation is formed. over damped damped t-o tan LLLLLLLLL Shot on OnePlus By Aastha LA