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. `0.693/b`
B. both kinetic and potential energies are maximum
C. `1/b`
D. `2/b`

Answer

D

Explanation

For damped harmonic motion,
or ma+m bv+kx=0
Solution of above equation is `x=A_(0)ebt//2 sin omega t,` with `omega^(2)=k//m-b^(2)/4`
where, amplitude drops exponentially with time.
i.e., `A_(tau)=A_(0)eb_(tau)//(2)`
Average time is that duration when amplitude drops by 63%, i.e., becomes `A_(0)//e`.
Thus , `A_(tau)=A_(0)//e=A_(0)eb_(tau)//2` or `b_(tau)//2=1 ` or `tau=2//b`