If a simple pendulum has significant amplitude (up to a factor of1//e of original) only in the period between `t-0s to t=tau s`, then `tau` may be called the average life of the pendulum. When the sphetical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with b as the constant of propotional to 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

a = retardation of pendulum = bv

`theta` is the angular displacement of bob.
Retardation force `F_(b) = mbv`
Net torque on bob about axis of rotation.
`tau = mgl sin theta + F_(b)l`
`Ialpha = mgl sin theta + mbvl`
`alpha = (-g)/(l) sin theta + (bc)/(l)` r> For small damping, `sin ~~ theta`
` alpha = (-g)/(l) theta + (bv)/(l) rArr (d^(2))/(dt^(2)) = (-g^(theta))/(l) + (bv)/(l)`
Solution of this differential equation
`theta = theta_(0)e^((-bt)/(2)) sin (omegat + phi)`
Angular amplitude ` = theta_(0)e^((-bt)/(2))`
Given, `(theta_(0))/(e) = theta_(0)e^((-bt)/(2)) " at" t = tau`
`e^(-1) = e^((-bt)/(2))`
`tau = (2)/(b)`