A man transforms into a giant such that all his linear dimensions become \(9\) times their original values. Assuming his density remains unchanged, the stress on his legs changes by a factor of:

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. \(\dfrac{mg}{Ka}\)

A boy’s catapult is made of rubber cord which is \(42~\text{cm}\) long, with \(6~\text{mm}\) diameter of a cross-section and of negligible mass. The boy keeps a stone weighing \(0.02~\text{kg}\) on it and stretches the cord by \(20~\text{cm}\) by applying a constant force. When released, the stone flies off with a velocity of \(20~\text{ms}^{-1}.\) Neglect the change in the area of cross-section of the cord while stretched. The Young’s modulus of rubber is closest to:
1. \( 10^3 ~\text{Nm}^{-2} \)
2. \(10^4~\text{Nm}^{-2} \)
3. \( 10^6 ~\text{Nm}^{-2} \)
4. \( 10^8~\text{Nm}^{-2} \) 

Young's moduli of two wires \(A\) and \(B\) are in the ratio \(10:4\). Wire \(A\) is \(2~\text{m}\) long and has radius \(R\). Wire \(B\) is \(1.6~\text{m}\) long and has radius \(2~\text{mm}\). If the two wires stretch by the same length for a given load, then the value of \(R\) is close to:
1. \(\sqrt{2} ~\text{mm} \) 
2. \(\frac {1} {\sqrt{2}}~\text{mm} \) 
3. \(2\sqrt{2} ~\text{mm} \) 
4. \(2~\text{mm} \)

In an experiment, brass and steel wires of length \(1~\text{m}\) each with areas of cross section \(1~\text{mm}^2\) are used. The wires are connected in series and one end of the combined wire is connected to a rigid support and other end is subjected to elongation. The stress required to produce a net elongation of \(0.2~\text{mm}\) is, [Given, the Young's Modulus for steel and brass are, respectively, \(120 \times 10^9 ~\text{N/m}^2\) and \(60 \times 10^9 ~\text{N/m}^2\)]
1. \( 4.0 \times 10^6 ~\text{N/m}^2\)
2. \( 1.2 \times 10^6~\text{N/m}^2\)
3. \( 1.8 \times 10^6~\text{N/m}^2\)
4. \(8 \times 10^6~\text{N/m}^2\)

The elastic limit of brass is 400 MPa. What should be the minimum diameter of a brass rod if it is to support a 400\(\pi \) N load without exceeding its elastic limit?
1. 1 mm
2. 1.5 mm
3. 2 mm
4. 2.5 mm

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. \(1.67\)

A uniform metallic wire is elongated by \(0.04\) m when subjected to a linear force \(F\). The elongation, if its length and diameter are doubled and subjected to the same force will be: