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An object wei...

Question

An object weights $m_{1}$ in a liquid of density $d_{1}$ and that in liquid of density $d_{2}$ is $m_{2}$ . The density $d$ of the object is

\frac{m_{2} d_{2} - m_{1} d_{1}}{m_{2} - m_{1}}

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\frac{m_{1} d_{1} - m_{2} d_{2}}{m_{2} - m_{1}}

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\frac{m_{2} d_{1} - m_{1} d_{2}}{m_{1} - m_{2}}

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\frac{m_{1} d_{2} - m_{2} d_{1}}{m_{1} - m_{2}}

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Solution

The correct option is D $\frac{m_{1} d_{2} - m_{2} d_{1}}{m_{1} - m_{2}}$
Let the mass and density of object in air be $M$ and $d$ respectively having volume $V$ .

Case (1): The object is immersed in liquid of density $d_{1}$
Thus the buoyancy force offered by liquid, $B = V d_{1} g$
$\Rightarrow$ $m_{1} = M g - V d_{1} g$
$\Rightarrow m_{1} = (V d) g - V d_{1} g$
$\Rightarrow m_{1} = (d - d_{1}) V g$ .............(1)

Case (2): The object is immersed in liquid of density $d_{2}$
Thus the buoyancy force offered by liquid, $B = V d_{2} g$
$\Rightarrow m_{2} = M g - V d_{2} g$
$\Rightarrow m_{2} = (V d) g - V d_{2} g$
$\Rightarrow m_{2} = (d - d_{2}) V g$ .............(2)

Dividing (1) and (2), we get:
$\Rightarrow \frac{m_{1}}{m_{2}} = \frac{d - d_{1}}{d - d_{2}}$

$\Rightarrow d = \frac{m_{1} d_{2} - m_{2} d_{1}}{m_{1} - m_{2}}$

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