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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 , let the velocity of the rod be v .
Applying Newtons law: F-iIB=ma…..(1)
Also `Biv=i_(1)R=(q)/( c)`
Applying Kcl, `i=i_(1)+(dq)/(dt)=(BlV)/( R)+(D)/(Dt)(BIvC) or i=(B l V)/( R)+B l C 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)(v)/(F-(B^(2)I^(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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