Which of the following different equations represents a damped harmonic oscillator ?

Options

A. `(d^(2)y)/(dt^(2))+y=0`
B. `(d^(2)y)/(d^(2))+gamma(dy)/(dt)+y=0`
C. `(d^(2)y)/(dt^(2))+k^(2)y=0`
D. `(dy)/(dt)+y=0`

Answer

B

Explanation

For a damped oscillator ` F prop v`
Total restoring force `F=-ky-bv`
b- damping constant, y- displacement
If `F-ky-bv`
then `(d^(2)y)/(dt^(2))=-ky-b(dy)/(dt)`
`m(d^(2)y)/(dt^(2))+ky+(dy)/(dt)=0`
`-: m` we get
`(d^(2)y)/(dt^(2))+(b)/(m)(dy)/(dt)+(k)/(m)y=0`
If `k=m and (b)/(m)=r`
Then the equation becomes
`(d^(2)y)/(dt^(2))+y(dy)/(dt)+y=0`