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Question

The extension in a uniform rod of length l, mass m, cross sectional area A and young's modulus Y when it is suspended vertically at one end is

A
mglAY
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B
mgl2AY
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C
2mglAY
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D
mgl4AY
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Solution

The correct option is B mgl2AY
Let us take a small part of rod of length dx at height x from lower end.


Tension at point P in the rod have to support the mass m of lower portion of rod below point P.
So, Tension at point P,
T=mg=mgxl .........(1)
[ Since rod is uniform ]
Now, extension of this small part of rod of width dx due to tension at P,
dl=TdxAY
dl=mgxdxAYl [ from (1) ]
Therefore, total extension in rod,
L=l0mgxdxYAl=mgl2AY

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