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If a simple pendulum has significant amp...

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:

A

`0.693/b`

B

both kinetic and potential energies are maximum

C

`1/b`

D

`2/b`

Text Solution

Verified by Experts

The correct Answer is:
D
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`
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