Question 153: Oscillations

Question

If a simple pendulum has significant amplitude (up to a factor of $$1/e$$ of original ) only in the period between $$t = 0s\,\,to\,\,t = 	au \,s,$$ then $$	au \,$$ 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 :

Accepted Answer

The equation of motion for the pendulum, suffering retardation
$$I\alpha  =  - mg\left( {\ell \sin 	heta } ight) - mbv\left( \ell  ight)$$ where $$I = m{\ell ^2}$$
and $$\alpha  = {d^2}	heta /d{t^2}$$
$$	herefore$$ $${{{d^2}	heta } \over {d{t^2}}} =  - {g \over \ell }	an 	heta  + {{bv} \over \ell }$$
on solving we get $$	heta  = {	heta _0}\,{e^{{{bt} \over 2}\sin \left( {\omega t + \phi } ight)}}$$
According to questions $${{{	heta _0}} \over e} = {	heta _0}{e^{{{ - b	au } \over 2}}}$$
$$	herefore$$ $$	au  = {2 \over b}$$

Answer

$${{0.693} \over b}$$

Answer

$$b$$

Answer

$${1 \over b}$$

Answer

$${2 \over b}$$

Oscillations

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