A solid sphere of density rho_s which is eta | vedclass

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(D) Let $V$ be the volume of the sphere,$\rho_s$ be the density of the sphere,and $\rho_w$ be the density of water.Given that the sphere is $\eta$ tim

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$(\eta - 1) mg$

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(D) Let $V$ be the volume of the sphere,$\rho_s$ be the density of the sphere,and $\rho_w$ be the density of water.Given that the sphere is $\eta$ times lighter than water,we have $\rho_w = \eta \rho_s$.The mass of the sphere is $m = V \rho_s$,so $V = \frac{m}{\rho_s}$.The forces acting on the sphere are:$1$. Weight of the sphere $(mg)$ acting downwards.$2$. Tension in the string $(T)$ acting downwards.$3$. Buoyant force $(F_B)$ acting upwards,where $F_B = V \rho_w g$.For equilibrium,the net force is zero: $F_B = T + mg$.Therefore,$T = F_B - mg = V \rho_w g - V \rho_s g$.Substituting $\rho_w = \eta \rho_s$ and $V \rho_s = m$:$T = V (\eta \rho_s) g - m g = \eta (V \rho_s) g - mg = \eta mg - mg = (\eta - 1) mg$.

$A$ solid sphere of density $\rho_s$ which is $\eta$ $( > 1)$ times lighter than water (density $\rho_w$) is suspended in a water tank by a string tied to its base as shown in the figure. If the mass of the sphere is $m$,then the tension in the string is given by:

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