(B) The thermal expansion of the rod if it were free to expand is given by $\Delta L = \alpha L T$.To prevent this expansion,a compressive force $F$ m

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$\frac{Y A \alpha T}{(1+\alpha T)}$

(B) The thermal expansion of the rod if it were free to expand is given by $\Delta L = \alpha L T$.To prevent this expansion,a compressive force $F$ must be applied.The stress produced is $\sigma = \frac{F}{A}$.The strain produced is $\epsilon = \frac{\Delta L}{L_{new}}$,where $L_{new} = L(1 + \alpha T)$.Using Young's modulus $Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L / L(1 + \alpha T)}$.Rearranging for $F$,we get $F = \frac{Y A \Delta L}{L(1 + \alpha T)}$.Substituting $\Delta L = \alpha L T$,we get $F = \frac{Y A (\alpha L T)}{L(1 + \alpha T)}$.Therefore,$F = \frac{Y A \alpha T}{1 + \alpha T}$.

Class 11 Physics Mechanical Properties of Solids Young’s Modulus

Medium

$A$ metal rod of length $L$ and cross-sectional area $A$ is heated through $T^{\circ} C$. What is the force required to prevent the expansion of the rod lengthwise? $[Y=$ Young's modulus of the material of rod,$\alpha=$ coefficient of linear expansion $]$

MHT CET 2017 All MHT CET

A
$\frac{Y A \alpha T}{(1-\alpha T)}$
B
$\frac{Y A \alpha T}{(1+\alpha T)}$
C
$\frac{(1-\alpha T)}{Y A \alpha T}$
D
$\frac{(1+\alpha T)}{Y A \alpha T}$

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Mechanical Properties of Solids

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Young’s Modulus

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$A$ metal rod of length $L$ and cross-sectional area $A$ is heated through $T^{\circ} C$. What is the force required to prevent the expansion of the rod lengthwise? $[Y=$ Young's modulus of the material of rod,$\alpha=$ coefficient of linear expansion $]$

Similar Questions

If the ratio of diameters,lengths,and Young's modulus of steel and copper wires shown in the figure are $p, q$ and $s$ respectively,then the corresponding ratio of increase in their lengths would be

$A$ metal rod of Young's modulus $Y$ and coefficient of thermal expansion $\alpha$ is held at its two ends such that its length remains invariant. If its temperature is raised by $t^{\circ} C$,the linear stress developed in it is

$A$ pendulum made of a uniform wire of cross-sectional area $A$ has a time period $T$. When an additional mass $M$ is added to its bob,the time period changes to $T_M$. If the Young's modulus of the material of the wire is $Y$,then $\frac{1}{Y}$ is equal to ($g$ = gravitational acceleration).

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