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Mechanical Properties Of Fluids

Question
CBSEENPH11020509

Two non-mixing liquids of densities straight rho and nstraight rho (n >1) are put in a container. The height of each liquid is h. a solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length pL (p<1) in the denser liquid. The density d is equal to,

  • open curly brackets 2 plus left parenthesis straight n plus 1 right parenthesis straight p close curly brackets rho
  • open curly brackets 2 plus left parenthesis straight n minus 1 right parenthesis straight p close curly brackets rho
  • {1+(n-1)p}straight rho

  • {1+(n+1)p}straight rho

Solution

C.

{1+(n-1)p}straight rho

The situation can be depicted as follows:

According to Archimedes principle,
Weight of the cylinder = (upthrust)1 + (upthrust)2
i.e.,
space space space space ALdg space equals space left parenthesis 1 minus straight P right parenthesis LAρg space plus space left parenthesis PLA right parenthesis nρg

rightwards double arrow space straight d space equals space left parenthesis 1 minus straight P right parenthesis straight rho space plus space Pnρ
space space space space space space space space space equals space straight rho space minus space Pρ space plus space nPρ
space space space space space space space space space equals space straight rho space plus space left parenthesis straight n minus 1 right parenthesis Pρ
space space space space space space space space space equals straight rho space left square bracket space 1 plus left parenthesis straight n minus 1 right parenthesis straight rho right square bracket