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JEE Main & Advanced Physics Wave Mechanics Question Bank Self Evaluation Test - Oscillations

  • question_answer
    If a simple pendulum has significant amplitude (upto a factor of \[1/e\] 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 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 the small) in seconds        

    A) \[\frac{0.693}{b}\]

    B) b

    C) \[\frac{1}{b}\]

    D) \[\frac{2}{b}\]

    Correct Answer: D

    Solution :

    [d] For damped harmonic motion, \[ma=-kx-mbv\] or \[ma+mbv+kx=0\] Solution to above equation is \[x={{A}_{0}}{{e}^{-\frac{bt}{2}}}\,\sin \,\omega t;\] with \[{{\omega }^{2}}=\frac{k}{m}-\frac{{{b}^{2}}}{4}\] where amplitude drops exponentially with time i.e., \[{{A}_{\tau }}={{A}_{0}}{{e}^{-\frac{b\tau }{2}}}\] Average time \[\tau \] is that duration when amplitude drops by 63%, i.e., becomes\[{{A}_{0}}/e\]. Thus, \[{{A}_{\tau }}=\frac{{{A}_{0}}}{e}={{A}_{0}}{{e}^{-\frac{b\tau }{2}}}\] or \[\frac{b\tau }{2}=1\,\,\,or\,\,\,\tau =\frac{2}{b}\]


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