A uniform rod of mass m , length L , area of cross-section A and Young's modulus Y hangs from ce

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8.Mechanical Properties of Solids

hard

A uniform rod of mass $m$, length $L$, area of cross-section $A$ and Young's modulus $Y$ hangs from the ceiling. Its elongation under its own weight will be

Zero

$\frac {mgL}{2AY}$

$\frac {mgL}{AY}$

$\frac {2mgL}{AY}$

Solution

Mass of section $\mathrm{BC}=\frac{\mathrm{M}}{\mathrm{L}}(\mathrm{L}-\mathrm{y})$

$\text { Tension at } \mathrm{B}=\mathrm{T}=\frac{\mathrm{m}}{\mathrm{L}}(\mathrm{L}-\mathrm{y}) \mathrm{g}$

$\therefore$ Elongation of element dy at $\mathrm{B}$.

$\mathrm{dx}=\mathrm{dy} \frac{\mathrm{T}}{\mathrm{AY}}=\frac{\mathrm{m}}{\mathrm{L}}(\mathrm{L}-\mathrm{y}) \mathrm{g} \frac{\mathrm{dy}}{\mathrm{AY}}$

Total elongation

$\mathrm{x}=\int \mathrm{dx}=\frac{\mathrm{mg}}{\mathrm{LAY}} \int_{0}^{\mathrm{L}}(\mathrm{L}-\mathrm{y}) \mathrm{dy}=\frac{\mathrm{mgL}}{2 \mathrm{YA}}$

Standard 11

Physics

The length of an elastic string is a metre when the longitudinal tension is $4\, N$ and $b$ metre when the longitudinal tension is $5\, N$. The length of the string in metre when the longitudinal tension is $9\, N$ is

hard

A beam of metal supported at the two ends is loaded at the centre. The depression at the centre is proportional to

medium

The temperature of a wire of length $1$ metre and area of cross-section $1\,c{m^2}$ is increased from $0°C$ to $100°C$. If the rod is not allowed to increase in length, the force required will be $(\alpha = {10^{ – 5}}/^\circ C$ and $Y = {10^{11}}\,N/{m^2})$

medium

Column$-II$ is related to Column$-I$. Join them appropriately :

Column $-I$ Column $-II$

$(a)$ When temperature raised Young’s modulus of body $(i)$ Zero

$(b)$ Young’s modulus for air $(ii)$ Infinite

$(iii)$ Decreases

$(iv)$Increases

medium

The length of wire, when $M_1$ is hung from it, is $I_1$ and is $I_2$ with both $M_1$ and $M_2$ hanging. The natural length of wire is ……..

hard

Column $-I$	Column $-II$
$(a)$ When temperature raised Young’s modulus of body	$(i)$ Zero
$(b)$ Young’s modulus for air	$(ii)$ Infinite
	$(iii)$ Decreases
	$(iv)$Increases

A uniform rod of mass $m$, length $L$, area of cross-section $A$ and Young's modulus $Y$ hangs from the ceiling. Its elongation under its own weight will be

Solution

Similar Questions

The length of an elastic string is a metre when the longitudinal tension is $4\, N$ and $b$ metre when the longitudinal tension is $5\, N$. The length of the string in metre when the longitudinal tension is $9\, N$ is

A beam of metal supported at the two ends is loaded at the centre. The depression at the centre is proportional to

The temperature of a wire of length $1$ metre and area of cross-section $1\,c{m^2}$ is increased from $0°C$ to $100°C$. If the rod is not allowed to increase in length, the force required will be $(\alpha = {10^{ – 5}}/^\circ C$ and $Y = {10^{11}}\,N/{m^2})$

Column$-II$ is related to Column$-I$. Join them appropriately : Column $-I$ Column $-II$ $(a)$ When temperature raised Young’s modulus of body $(i)$ Zero $(b)$ Young’s modulus for air $(ii)$ Infinite $(iii)$ Decreases $(iv)$Increases

The length of wire, when $M_1$ is hung from it, is $I_1$ and is $I_2$ with both $M_1$ and $M_2$ hanging. The natural length of wire is ……..

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Column$-II$ is related to Column$-I$. Join them appropriately :

Column $-I$ Column $-II$

$(a)$ When temperature raised Young’s modulus of body $(i)$ Zero

$(b)$ Young’s modulus for air $(ii)$ Infinite

$(iii)$ Decreases

$(iv)$Increases