If a simple pendulum has significant amplitude (up to a factor of `(1)/(e)` of orginal) only in the period between `t = 0` sec to `t = tau` sec, then `tau` may be called the average life of pendulum. When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity, with 'b' as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds:

Options

A. `(2)/(b)`
B. `(0.693)/(b)`
C. `b`
D. `1/b`

Answer

A

Explanation

Equation of damped simple pendulum
`(d^(2)x)/(dt^(2)) = -bv + g sin theta`
`rArr (d^(2)x)/(dt^(2)) = - bv + (g)/(l) x = 0`
By solving above equation
`x = A_(0)e^(-(b)/(2)l) sin (omegat + phi)`
At `t = tau, A == (A_(0))/(2)` so `tau = (2)/(b)`