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 Bmgl2AY 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=m′g=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=l∫0mgxdxYAl=mgl2AY