A body of density ${d_1}$ is counterpoised by $Mg$ of weights of density ${d_2}$ in air of density $d$. Then the true mass of the body is
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(d)Let ${M_0} = $mass of body in vacuum.
Apparent weight of the body in air = Apparent weight of standard weights in air
$ \Rightarrow $ Actual weight -upthrust due to displaced air
= Actual weight -upthrust due to displaced air
$ \Rightarrow $${M_0}g - \left( {\frac{{{M_0}}}{{{d_1}}}} \right)\,dg = Mg - \left( {\frac{M}{{{d_2}}}} \right)\,dg$$ \Rightarrow {M_0} = \frac{{M\left[ {1 - \frac{d}{{{d_2}}}} \right]}}{{\left[ {1 - \frac{d}{{{d_1}}}} \right]}}$
Apparent weight of the body in air = Apparent weight of standard weights in air
$ \Rightarrow $ Actual weight -upthrust due to displaced air
= Actual weight -upthrust due to displaced air
$ \Rightarrow $${M_0}g - \left( {\frac{{{M_0}}}{{{d_1}}}} \right)\,dg = Mg - \left( {\frac{M}{{{d_2}}}} \right)\,dg$$ \Rightarrow {M_0} = \frac{{M\left[ {1 - \frac{d}{{{d_2}}}} \right]}}{{\left[ {1 - \frac{d}{{{d_1}}}} \right]}}$
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