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A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere,

\left ( \frac{dr}{r} \right ), is:

 

  • Option 1)

    \frac{mg}{Ka}

  • Option 2)

    \frac{Ka}{mg}

  • Option 3)

    \frac{Ka}{3mg}

  • Option 4)

    \frac{mg}{3 Ka}

 

Answers (2)

best_answer

As we learnt that

 

Bulk Modulus -

Ratio of normal stress to volumetric strain.

K=\frac{f/A}{-\Delta v/v}=\frac{-Fv}{A\Delta v}

K=\frac{-Pv}{\Delta v}

v = Original  volume

\Delta v = Change in volume

P = Increase in pressure

-ve(sign) shows volume (\Delta v) decrease.

- wherein

 

 \Delta P=\frac{mg}{a}\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: V= \frac{4\pi }{3}r^{3}

K=-\frac{\Delta P}{(\frac{\Delta V}{V})}\: \: \: \: \: \: \: \: \: \: \:\: \: \: \therefore \frac{dV}{V}=3.\frac{dr}{r}

K=\frac{-\Delta P}{3.(\frac{dr}{r})}

or\: \frac{dr}{r}=\frac{-mg}{3Ka}


Option 1)

\frac{mg}{Ka}

This is incorrect

Option 2)

\frac{Ka}{mg}

This is incorrect

Option 3)

\frac{Ka}{3mg}

This is incorrect

Option 4)

\frac{mg}{3 Ka}

This is correct

Posted by

Aadil

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