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Find the velocity of the moving rod at t...

Find the velocity of the moving rod at time `t` if the initial velocity of the rod is `v` and `a` constant force `F` is applied on the rod.Neglect the resistance of the rod.

Text Solution

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At any time `t`, let the velocity of the rod be `v`.
Applying Newtons law:`F-ilB=ma` ..(1)
Also `Blv=i_(1),R=q/c`
Applying `Kcl`,
`i=i_(1)+(dq)/(dt)=(BlV)/R+d/(dt)(BlvC)`
or `i=(BlV)/R+BlC a`
Putting the value of `i` in eq.(1), `F-(B^(2)L^(2)V)/R=(m+B^(2)L^(2)C)a=(m+B^(2)L^(2)C)(dv)/(dt)`
`(m+B^(2)L^(2)C)(dv)/(F-(B^(2)L^(2)V)/R)=dt`
Integrating both sides, and solving we get
`v=(FR)/(B^(2)l^(2))(1-e(tB^(2)l^(2))/(R(m+CB^(2)l^(2))))`
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