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A body of mass m starts moving from rest...

A body of mass m starts moving from rest along x-axis so that its velocity varies as `v=asqrts` where is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

Text Solution

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Given, velocity `v = alpha s^(1//2)`
Acceleration, `a = (dv)/(dt) = (d)/(dt) (alpha s^(1//2)) = (1)/(2) alpha s^(-1//2) ((ds)/(dt)) = (1)/(2) alpha s^(-1//2) v = (1)/(2) alpha s^(-1//2) (alpha s^(1//2)) = (alpha^(2))/(2)`
Now force, `F = ma = (m alpha^(2))/(2)`.
Distance covered by locomotive in first t second
`s = ut + (1)/(2) at^(2) = 0 + (1)/(2) ((alpha^(2))/(2))t^(2) = (alpha^(2)t^(2))/(4)`, Work done, `W = Fs = (m alpha^(2))/(2) xx (alpha^(2)t^(2))/(4) = (m alpha^(4) t^(2))/(8)`.
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