Two simple harmonic motions are represented by the equations ${y_1} = 0.1\sin \left( {100\pi t + \frac{\pi }{3}} \right)$ and ${y_2} = 0.1\cos \pi t.$ The phase difference of the velocity of particle $1$ with respect to the velocity of particle $2$ is
AIEEE 2005, Medium
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(c) ${v_1} = \frac{{d{y_1}}}{{dt}} = 0.1 \times 100\pi \cos \left( {100\pi t + \frac{\pi }{3}} \right)$
${v_2} = \frac{{d{y_2}}}{{dt}} = - 0.1\pi \sin \pi t = 0.1\pi \cos \left( {\pi t + \frac{\pi }{2}} \right)$
Phase difference of velocity of first particle with respect to the velocity of $2^{nd}$ particle at $t = 0$ is
$\Delta \phi = {\phi _1} - {\phi _2} = \frac{\pi }{3} - \frac{\pi }{2} = - \frac{\pi }{6}.$
${v_2} = \frac{{d{y_2}}}{{dt}} = - 0.1\pi \sin \pi t = 0.1\pi \cos \left( {\pi t + \frac{\pi }{2}} \right)$
Phase difference of velocity of first particle with respect to the velocity of $2^{nd}$ particle at $t = 0$ is
$\Delta \phi = {\phi _1} - {\phi _2} = \frac{\pi }{3} - \frac{\pi }{2} = - \frac{\pi }{6}.$
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