If a simple pendulum has significant amplitude (up to a factor of 1/e of original) only in the period between t = Os to `t = tau s`, then `tau` may be called the average life of the 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. (a) (0.693)/b`
B. (b) `b`
C. (c )`1/b`
D. (d) `2/b`

Answer

D

Explanation

(d) The equation of motion for the pendulum, suffering retardation
`Ia=-mg(lsin theta)-mbv(l)where I=ml^(2)`
and `a=d^(2)theta//dt^(2)`
:. (d^(2)theta)/(dt^(2)=g/ls tan theta+(dv)/l`
On solving we get `theta_(0)/e=theta_(0) e^(-(c=bt)/2sin(omegat+phi).
According to questons (theta_(0)/e=theta_(0)e^(-bphi)/2`
:. phi=2/b`.