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(C) In the frame of the sphere,the liquid experiences a pseudo-force in the backward direction with an acceleration $a = 2g$. The effective accelerati

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$P_0 + \rho gR\sqrt{5}$

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(C) In the frame of the sphere,the liquid experiences a pseudo-force in the backward direction with an acceleration $a = 2g$. The effective acceleration vector $\vec{g}_{eff}$ is the vector sum of the acceleration due to gravity $\vec{g}$ (acting downwards) and the pseudo-acceleration $\vec{a}_{pseudo} = -2g\hat{i}$ (acting backwards).Thus,$\vec{g}_{eff} = -2g\hat{i} - g\hat{j}$.The magnitude of the effective acceleration is $g_{eff} = \sqrt{(2g)^2 + g^2} = \sqrt{5g^2} = g\sqrt{5}$.The pressure in the liquid varies as $P = P_{min} + ho \vec{g}_{eff} \cdot \vec{r}$,where $\vec{r}$ is the displacement vector from the point of minimum pressure to the point of interest.The minimum pressure occurs at the point furthest in the direction of $\vec{g}_{eff}$. The distance from the centre to this point is $R$.The pressure at the centre is $P_c = P_{min} + ho g_{eff} 	imes R$.Substituting the values,$P_c = P_0 + ho (g\sqrt{5}) R = P_0 + ho gR\sqrt{5}$.

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