First slide

Damped and forced oscillations

Question

Which of the following figure represents damped harmonic motion

Moderate

A

1 & 2
B

2 & 3
C

1,2,3 & 4
D

2 & 4

Solution

$d i f f e r e n t i a l e q u a t i o n o f d a m p e d h a r m o n i c o s c i l l a t o r i s \frac{d^{2} x}{d t^{2}} + 2 b \frac{d x}{d t} + ω^{2} x = 0 w h e r e x i s d i s p l a c e m e n t o f o s c i l l a t o r o p t i o n 2 r e p r e s e n t s u n d e r d a m p e d m o t i o n, w h e n b^{2} < ω^{2}, h e r e \sqrt{b^{2} - ω^{2}} i s i m a g i n a r y, a n d a m p l i t u d e = a e^{- b t}, t i m e p e r i o d T = \frac{2 π}{\sqrt{ω^{2} - b^{2}}} o p t i o n 4 r e p r e s e n t s o v e r d a m p e d m o t i o n, w h e n b^{2} > ω^{2}, h e r e \sqrt{b^{2} - ω^{2}} i s r e a l$

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