A disc of radius R and mass M having a point charge -Q at the centre is kept on a plane. At a distance ' I vertically above from the centre of the disc, another charge +Q is kept. The disc is sli

Question

A disc of radius R and mass M having a point charge -Q at the centre is kept on a plane. At a distance ' I vertically above from the centre of the disc, another charge +Q is kept. The disc is slightly displaced in the horizontal direction. It rolls without slipping. For very small displacement time period of oscillation is.

TestHub · Accepted Answer

For a small horizontal shift x the attraction between the charges gives a restoring horizontal force $F = \frac{k Q ^{2}}{l ^{3}} x$ (to leading order).

Rolling without slipping: $K E = \frac{1}{2} M \overset{x}{˙}^{2} + \frac{1}{2} (\frac{1}{2} M R^{2}) (\frac{x ˙}{R})^{2} = \frac{3}{4} M \overset{x}{˙}^{2}$ .

$ω^{2} = \frac{k Q ^{2} / l ^{3}}{3 M /2} = \frac{2 k Q ^{2}}{3 M l ^{3}}$ ⇒ $T = 2 π \frac{3 M l ^{3}}{2 k Q ^{2}} = \frac{2 π}{Q} 6 π ε_{0} M l^{3}$ - (A)

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