Question 511: Waves

Question

A point performs simple harmonic oscillation of period T and the equation of motion is given by x = a sin($$\omega $$t + $$\pi $$/6). After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

Accepted Answer

We have $$x = a\sin \left( {\omega t + {\pi  \over 6}} ight)$$
$$ 	herefore $$ Velocity, $$v = {{dx} \over {dt}} = a\omega \cos \left( {\omega t + {\pi  \over 6}} ight)$$
Maximum velocity = a$$\omega $$
According to question,
$${{a\omega } \over 2} = a\omega \cos \left( {\omega t + {\pi  \over 6}} ight)$$
$$ \Rightarrow \cos \left( {\omega t + {\pi  \over 6}} ight) = {1 \over 2} = \cos {60^o}$$ or $${\cos {\pi  \over 3}}$$
$$ \Rightarrow \omega t = {\pi  \over 3} - {\pi  \over 6} \Rightarrow \omega t = {\pi  \over 6}$$
$$ \Rightarrow {{2\pi } \over T}.t = {\pi  \over 6} \Rightarrow t = {T \over {12}}$$

Answer

T/3

Answer

T/12

Answer

T/8

Answer

T/6

Waves

A point performs simple harmonic oscillation of period T and the equation of motion is given by x = a sin($\omega t + \pi $/6). After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

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