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(D) The gravitational potential $V$ at a distance $r = \frac{R}{2}$ inside a uniform sphere of mass $M$ and radius $R$ is given by:$V = -\frac{GM}{2R^

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$\sqrt{\frac{11GM}{4R}}$

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(D) The gravitational potential $V$ at a distance $r = \frac{R}{2}$ inside a uniform sphere of mass $M$ and radius $R$ is given by:$V = -\frac{GM}{2R^3} (3R^2 - r^2)$Substituting $r = \frac{R}{2}$:$V = -\frac{GM}{2R^3} (3R^2 - \frac{R^2}{4}) = -\frac{GM}{2R^3} (\frac{11R^2}{4}) = -\frac{11GM}{8R}$The total energy of the particle of mass $m_0$ at this position is $E_i = K_i + U_i = \frac{1}{2}m_0 v_e^2 + m_0 V$.For the particle to escape,the final total energy $E_f$ at infinity must be at least $0$.$E_i = E_f \implies \frac{1}{2}m_0 v_e^2 - \frac{11GMm_0}{8R} = 0$$\frac{1}{2} v_e^2 = \frac{11GM}{8R}$$v_e^2 = \frac{11GM}{4R}$$v_e = \sqrt{\frac{11GM}{4R}}$

$A$ very small groove is made in the earth,and a particle of mass $m_0$ is placed at $\frac{R}{2}$ distance from the centre. Find the escape speed of the particle from that place.

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