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A particular moving in a straight line h...

A particular moving in a straight line has velocity v give by `v^(2)=alpha-betay^(2)` where `alpha and beta` are constant and y is its distance from a fixed point in the line . Show that the motion of the particle is SHM . Find its time period and amplitude.

Text Solution

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`v^(2)=alpha -betay^(2)" "…(1)`
Differentiating it with respect to time , t we have
`2v(dv)/(dt) =-beta^(2)y(dv)/(dt)" "…(2)`
As a and y have negative sign show that acceleration is directed towards mean position So, if the particle is left force , it will executes S.H.M
`w^(2)=beta`
`w=sqrt(beta)`
`:.` Time period , T `=(2pi)/(w) =(2pi)/(sqrt(beta)`
We know that , v= 0 , when y = r form equation (2).
`0=alpha -betar^(2)`
`A=sqrt((alpha)/(beta))`
amplitude , `A=sqrt((alpha)/(beta))`
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