Q. No.A man grows into a giant such that his linear dimensions increase by a factor of \(9.\) Assuming that his density remains the same, the stress in the leg will change by a factor of:
1. \(9\)
2. \(\dfrac{1}{9}\)
3. \(81\)
4. \(\dfrac{1}{81}\)
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, \( \left (\frac{dr}{r} \right )\) is:
1. \(\frac{Ka}{mg}\)
2. \(\frac{Ka}{3mg}\)
3. \(\frac{mg}{3Ka}\)
4. \(\frac{mg}{Ka}\)
| 1. | \(5.2\times10^{6}~\text{N/m}^{2}\) | 2. | \(6.2\times10^{6}~\text{N/m}^{2}\) |
| 3. | \(4.8\times10^{6}~\text{N/m}^{2}\) | 4. | \(3.1\times10^{6}~\text{N/m}^{2}\) |
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
(Breaking stress of wire = \(4.8 \times 10^7 ~\text{Nm}^{-2}\) and area of cross-section of the wire = \(10^{-2}~ \text {cm}^{-2}\))
1. \(4~\text{rad s}^{-1}\)
2. \(6~\text{rad s}^{-1}\)
3. \(8~\text{rad s}^{-1}\)
4. \(9~\text{rad s}^{-1}\)
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:
| 1. | \(1\) cm | 2. | \(2 \) cm |
| 3. | \(3\) cm | 4. | \(6\) cm |
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