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A metal wire of uniform cross-sectional ...

A metal wire of uniform cross-sectional area A and length L, has mass m. It is suspended vertically from a ceiling. If its Young's modulus is Y, then the elongation `Deltal` of wire due to its own weight will be .........

A

zero

B

`(mgL)/(2AY)`

C

`(mgL)/(AY)`

D

`(2mgL)/(AY)`

Text Solution

Verified by Experts

The correct Answer is:
B
Let us consider an element length dx. Now consider small element from a support (celling) at x distance. The length of wire below this element `= L - x`
Hence the tension at this point = weight of wire below this point
`therefore T = ((L -x )W)/(L)`
The elongation in this element
`= ("original length" xx "stress")/(Y)`
`= (Tdx)/(AY) = ((L -x ) Edx)/(LAY)`
`therefore ` Total elongation in length `= int _(0) ^(L) ((L -x) W dx )/( LAY) = (W)/( LAY) int _(0) ^(L) (L-x) dx`
`= (W)/( LAY) [ Lx - (x ^(2))/(2) ] _(0) ^(L) = (WL )/( 2 AY) = (mgL)/(2 AY)`
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