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Q. No.A uniform rod \(AB\) is rotated at a constant angular speed about its end \(A,\) the rotation axis being perpendicular to \(AB.\) During rotation, stresses are set up in the rod. Let the stress at \(A\) be \(\sigma_A\) and that at the centre \(C\) be \(\sigma_C.\) Then:
1. \(\sigma_A=\sigma_C\)
2. \(\sigma_A=2\sigma_C\)
3. \(\sigma_C=2\sigma_A\)
4. \(\sigma_C=\dfrac34\sigma_A\)
Subtopic: Stress - Strain |
Level 4: Below 35%
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An ideal gas has an isothermal bulk modulus of \(B_1.\) Its volume is doubled, isothermally. The bulk modulus is now:
| 1. | \(B_1\) | 2. | \(2B_1\) |
| 3. | \( \dfrac {B_1}{2}\) | 4. | \(\dfrac {B_1}{\sqrt 2}\) |
Subtopic: Shear and bulk modulus |
Level 3: 35%-60%
Hints
A large cylindrical piece of a dense solid elastic metal stands on its end as shown in the figure. The metal is uniform and isotropic. The stress in the material as a function of height is shown correctly by:
| 1. |  |
2. |  |
| 3. |  |
4. |  |
Subtopic: Stress - Strain Curve |
Level 3: 35%-60%
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An extremely long solid rod of length \(L\) starts falling longitudinally towards a large point mass \(M,\) the near end of the rod being at a distance \(L\) from the mass \(M.\)
The rod experiences:
The rod experiences:
| 1. | no stress | 2. | compressive stress |
| 3. | tensile stress | 4. | shear stress |
Subtopic: Stress - Strain |
Level 3: 35%-60%
Hints
A cube made of a homogeneous, isotropic elastic solid is acted upon by forces of equal magnitude acting perpendicular to its opposite faces as shown. Forces are applied uniformly over the area of each face. The stress at the centre of the cube is:
1. tensile
2. compressive
3. shear
4. zero
1. tensile2. compressive
3. shear
4. zero
Subtopic: Stress - Strain |
Level 3: 35%-60%
Hints
A simple pendulum is made with a thin wire (length: \(l,\) area: \(A,\) Young's modulus: \(Y\)) attached to a heavy bob of mass \(M.\) The pendulum is released from the rest with the bob at the same level as the point of suspension and swings down in a circular arc. The elongation in the wire when the bob reaches the lowest point is:
| 1. | \(\dfrac{3 M g l}{A Y}\) | 2. | \(\dfrac{2 M g l}{A Y}\) |
| 3. | \(\dfrac{3 M g l}{2 A Y}\) | 4. | \(\dfrac{M g l}{A Y}\) |
Subtopic: Hooke's Law |
Level 3: 35%-60%
Hints
A uniform rod of mass \(m,\) having cross-section \(A\) is pushed along its length \((L)\) by means of a force of magnitude, \(F.\) There is no friction anywhere. Ignore the weight of the rod. The longitudinal stress in the rod, at a distance \(\dfrac{L}{3}\) from the left end, is:
| 1. | tensile, \(\dfrac{F}{3A}\) | 2. | compressive, \(\dfrac{F}{3A}\) |
| 3. | tensile, \(\dfrac{2F}{3A}\) | 4. | compressive, \(\dfrac{2F}{3A}\) |
Subtopic: Stress - Strain |
Level 3: 35%-60%
Hints
A uniform rod of mass \(10~\text{kg}\) and length \(6~\text m\) is suspended vertically from the ceiling, as shown in the figure. The cross-sectional area of the rod is \(3~\text{mm}^2,\) and its Young’s modulus is \(2\times10^{11}~\text{N/m}^2.\) The extension in the length of the rod is: (take \(g=10~\text{m/s}^2\))
1. \(1~\text{mm}\)
2. \(0.5~\text{mm}\)
3. \(0.25~\text{mm}\)
4. \(1.2~\text{mm}\)
1. \(1~\text{mm}\)
2. \(0.5~\text{mm}\)
3. \(0.25~\text{mm}\)
4. \(1.2~\text{mm}\)
Subtopic: Elasticity |
53%
Level 3: 35%-60%
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Hints
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Choose the correct expression that relates Poisson’s ratio \(\sigma,\) bulk modulus \(B,\) and modulus of rigidity \(G.\)
1. \(\mathit{\sigma}{=}\dfrac{{3}{B}{-}{2}{G}}{{2}{G}{+}{6}{B}}\)
2. \(\mathit{\sigma}{=}\dfrac{{6}{B}{+}{2}{G}}{{3}{B}{-}{2}{G}}\)
3. \(\mathit{\sigma}{=}\dfrac{9BG}{{3}{B}{+}{G}}\)
4. \({B}{=}\dfrac{{3}\mathit{\sigma}{-}{3}{G}}{{6}\mathit{\sigma}{+}{2}{G}}\)
1. \(\mathit{\sigma}{=}\dfrac{{3}{B}{-}{2}{G}}{{2}{G}{+}{6}{B}}\)
2. \(\mathit{\sigma}{=}\dfrac{{6}{B}{+}{2}{G}}{{3}{B}{-}{2}{G}}\)
3. \(\mathit{\sigma}{=}\dfrac{9BG}{{3}{B}{+}{G}}\)
4. \({B}{=}\dfrac{{3}\mathit{\sigma}{-}{3}{G}}{{6}\mathit{\sigma}{+}{2}{G}}\)
Subtopic: Elasticity |
59%
Level 3: 35%-60%
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Two wires of identical dimensions but of different materials having Young's moduli \(Y_1, Y_2\) are joined end to end. When the first wire is under a tension \(T,\) it elongates by \(x_1\) while the second wire elongates by \(x_2\) under the same tension \(T.\) The elongation of the composite wire when it is under tension \(T\) is:
| 1. | \(x_1+x_2\) | 2. | \(\dfrac{Y_1x_1+Y_2x_2}{Y_1+Y_2}\) |
| 3. | \(\dfrac{x_1+x_2}{2}\) | 4. | \(\dfrac{Y_1x_2+Y_2x_1}{Y_1+Y_2}\) |
Subtopic: Young's modulus |
Level 3: 35%-60%
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