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Q. No.A metal rod of Young's modulus Y and coefficient of thermal expansion is held at its two ends such that its length remains invariant. If its temperature is raised by , the linear stress developed in it is:
1.
2.
3.
4.
Subtopic: Thermal Stress |
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A compressive force \(F\) is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by \(\Delta{T}.\) The net change in its length is zero. Let \(l\) be the length of the rod, \(A\) its area of cross-section, \(Y\) is Young's modulus and \(\alpha\) is coefficient of linear expansion. Then, the force \(F\) is equal to:
1. \(\frac{{AY}}{\alpha \Delta{T}}\)
2. \(\text {A}Y\alpha \Delta {T}\)
3. \(l^2 {Y}\alpha \Delta {T}\)
4. \(l {A}{Y} \alpha\Delta{T}\)
1. \(\frac{{AY}}{\alpha \Delta{T}}\)
2. \(\text {A}Y\alpha \Delta {T}\)
3. \(l^2 {Y}\alpha \Delta {T}\)
4. \(l {A}{Y} \alpha\Delta{T}\)
Subtopic: Thermal Stress |
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An external pressure \(P\) is applied on a cube at \(0^\circ\text{C}\) so that it is equally compressed from all sides. \(K\) is the bulk modulus of the material of the cube and \(\alpha\) is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:
1. \( \dfrac{{P}}{3 \alpha {K}}\)
2. \( \dfrac{{P}}{\alpha{K}} \)
3. \( \dfrac{3 \alpha}{{PK}} \)
4. \(3 {PK} \alpha\)
Subtopic: Thermal Stress |
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Two uniform rods, \(AB\) and \(BC,\) have Young's moduli of \(1.2~\times~ 10^{11}~ \text{N/m}^2\) and \(1.5~\times~ 10^{11}~ \text{N/m}^2,\) respectively. The coefficient of linear expansion for rod \(AB\) is \(1.5~\times~ 10^{-5} /^\circ \text{C}.\) If both rods have equal cross-sectional area, then the coefficient of linear expansion of \(BC,\) for which there is no shift of the junction at all temperatures, is:
| 1. | \(1.5~\times~ 10^{-5} /^\circ \text{C}\) | 2. | \(1.2~\times~ 10^{-5} /^\circ \text{C}\) |
| 3. | \(0.6~\times~ 10^{-5} /^\circ \text{C}\) | 4. | \(0.75~\times~ 10^{-5} /^\circ \text{C}\) |
Subtopic: Thermal Stress |
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A rod of length \({L}\) and uniform cross-sectional area \({A},\) made of a metal with coefficient of linear expansion \(\alpha,\) is held at both ends so that it cannot expand. When the temperature of the rod is increased by \(\Delta T,\) a compressive force \(F\) develops in the rod. What is the Young’s modulus \(Y\) of the metal?
| 1. | \( \dfrac{F} { {A} \alpha \Delta{T}}\) | 2. | \( \dfrac{F} { {A} \alpha (\Delta{T}-273)}\) |
| 3. | \( \dfrac{F} { 2{A} \alpha \Delta{T}}\) | 4. | \( \dfrac{2F} { {A} \alpha \Delta{T}}\) |
Subtopic: Thermal Stress |
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A steel rail of length \(5~\text{m}\) and cross-sectional area \(40~\text{cm}^2\) is rigidly fixed at both ends, preventing it from expanding when the temperature rises by \(10^\circ\text{C}.\) Given that the coefficient of linear expansion of steel is \(1.2\times10^{-5}~\text{K}^{-1} \) and its Young’s modulus is \(2\times10^{11}~\text{Nm}^{-2}, \) the approximate force developed in the rail is:
1. \(2\times10^{9}~\text{N} \)
2. \(3\times 10^{-5}~\text{N} \)
3. \(2\times10^{7}~\text{N} \)
4. \(1\times10^{5}~\text{N} \)
1. \(2\times10^{9}~\text{N} \)
2. \(3\times 10^{-5}~\text{N} \)
3. \(2\times10^{7}~\text{N} \)
4. \(1\times10^{5}~\text{N} \)
Subtopic: Thermal Stress |
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