A non conducting uniform thin shell of mass $m$ and radius $R$ has uniform charge density $+ σ$ on one half and $- σ$ on another half as shown in figure. It is placed on a rough non conducting horizontal plane. At t = 0 a uniform electric field $\vec{E} = E_{0} \hat{j} N / C$ is switched on and the solid sphere starts rolling without slipping . The speed of center of the sphere when it rotates through an angle of 53⁰ is $\sqrt{\frac{n π R^{3} σ_{0} E_{0}}{m}}$ . Find $n$ .

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answer is 1.92.

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Detailed Solution

As the work done by friction is zero,

$Δ K = - Δ U \Rightarrow \frac{1}{2} m V^{2} [1 + \frac{2}{3}] = - P E (\cos (\frac{π}{2} + 53) - \cos π / 2) = P E \times \frac{4}{5} = σ_{0} \times 2 π R^{2} \times R \times E_{0} \times \frac{4}{5} ∴ v = \sqrt{\frac{48 π R^{3} σ_{0} E_{0}}{25 m}}$