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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 the entire cross-section of the 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, \( \left (\dfrac{dr}{r} \right ) \) is:

1. \(\dfrac{Ka}{mg}\) 2. \(\dfrac{Ka}{3mg}\)
3. \(\dfrac{mg}{3Ka}\) 4. \(\dfrac{mg}{Ka}\)
Subtopic:  Shear and bulk modulus |
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A cube of metal is subjected to a hydrostatic pressure of \(4~\text{GPa}\). The percentage change in the length of the side of the cube is close to: (Given bulk modulus of metal, \(B=8 \times 10^{10} ~\text{Pa}\))
1. \(5\)
2. \(20\)
3. \(0.6\)
4. \(1.67\)

Subtopic:  Shear and bulk modulus |
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The normal density of a material is \(\rho\) and its bulk modulus of elasticity is \(K\). The magnitude of the increase in the density of material when a pressure \(P\) is applied uniformly on all sides, will be:

1. \(\dfrac{\rho K}{P}\) 2. \(\dfrac{\rho P}{K}\)
3. \(\dfrac{K}{P\rho}\) 4. \(\dfrac{PK}{\rho}\)
Subtopic:  Shear and bulk modulus |
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The bulk modulus of a liquid is \(3\times10^{10}\) Nm–2. The pressure required to reduce the volume of liquid by \(2\text{%}\) is:
1. \(3\times10^{8}\) Nm–2
2. \(9\times10^{8}\) Nm–2
3. \(6\times10^{8}\) Nm–2
4. \(12\times10^{8}\) Nm–2
Subtopic:  Shear and bulk modulus |
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A square aluminium (shear modulus is \(25\times10^{9}~\text{Nm}^{-2}\)) slab of side \(60~\text{cm}\) and thickness of \(15~\text{cm}\) is subjected to a shearing force (on its narrow face) of \(18.0\times10^{4}~\text {N}.\) The lower edge is riveted to the floor. The displacement of the upper edge is:
1. \(30~\mu \text m\) 
2. \(48~\mu \text m\) 
3. \(16~\mu \text m\) 
4. \(64~\mu \text m\) 
Subtopic:  Shear and bulk modulus |
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A sphere of small size is at the bottom of a lake of depth \(200 ~\text m.\) Due to pressure its fractional change in volume is \(\alpha\times 10^{-7}.\)  What is the value of \(\alpha,\) if the bulk modulus of the sphere is \(5 × 10^{12}~\text{Pa}?\) (use \(g = 10 ~\text{m/s}^2\) )
      
1. \(5\)
2. \(4\)
3. \(6\)
4. \(10\)
 
Subtopic:  Shear and bulk modulus |
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An object is located at \(2 ~\text{km}\) beneath the surface of the water. If the fractional compression \(\frac{\Delta{V}}{{V}}\) is \(1.36\%,\) the ratio of hydraulic stress to the corresponding hydraulic strain will be:
(Given: density of water is \(1000 ~\text{kgm}^{-3}\) and \(g = 9.8 ~\text{ms}^{-2}\))
1. \(1.44 \times 10^7 ~\text{Nm}^{-2}\)
2. \(1.96 \times 10^7~\text{Nm}^{-2}\)
3. \(2.26 \times 10^9~\text{Nm}^{-2}\)
4. \(1.44 \times 10^9~\text{Nm}^{-2}\)
Subtopic:  Shear and bulk modulus |
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At what depth in the deep sea will the volume of a rubber ball decrease by \(0.5\%?\)
(given that the bulk modulus of rubber is \(9.8 \times 10^8 ~\text{Nm}^{-2}\) and the density of seawater is \(10^3 ~\text{kg m}^{-3},~{g}=9.8~\text{m/s}^2)\)
1. \(700~\text m\)
2. \(600~\text m\)
3. \(500~\text m\)
4. \(400~\text m\)

 
Subtopic:  Shear and bulk modulus |
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The increase in pressure required to decrease the volume of a water sample by \(0.2\%\) is \(P×10^5~\text {NM}^{-2}.\) The bulk modulus of water is \(2.15 × 109~ \text { Nm}^ {-2}\). The value of \(P\) is:
1. \(79\)
2. \(43\)
3. \(28\)
4. \(134\)
Subtopic:  Shear and bulk modulus |
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The volume contraction of a solid copper cube of edge length \(10~\text{cm},\) when subjected to a hydraulic pressure of \(7 \times 10^6~\text{Pa} \) would be:
(Given the bulk modulus of copper \(=1.4 \times 10^{11} \mathrm{Nm}^{-2}\))
1. \(150~\text{mm}^3\)
2. \(50~\text{mm}^3\)
3. \(125~\text{mm}^3\)
4. \(108~\text{mm}^3\)
Subtopic:  Shear and bulk modulus |
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