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Mechanical Properties Of Fluids

Question
CBSEENPH11019977

A rigid bar of mass 15 kg is supported symmetrically by three wires each 2.0 m long. Those at each end are of copper and the middle one is of iron. Determine the ratio of their diameters if each is to have the same tension.

Solution

Tension force acting on each wire is the same. Hence, the extension is the same for each wire. 
Since, the wires are of the same length, the strain will also be same.
Young's modulus is given by,
 straight Y space equals space Stress over Strain space equals space fraction numerator begin display style bevelled straight F over straight A end style over denominator Strain end fraction space equals space fraction numerator begin display style bevelled fraction numerator 4 straight F over denominator πd squared end fraction end style over denominator Strain end fraction space space space space space space space space space space... space left parenthesis 1 right parenthesis

where comma space
straight F space equals space Tension space force
straight A space equals space Area space of space cross minus section
space straight d space equals space Diameter space of space the space wire

From space equation space left parenthesis straight i right parenthesis comma space we space have

straight Y proportional to open parentheses 1 over straight d squared close parentheses

Young apostrophe straight s space modulus space for space iron comma space straight Y subscript 1 space equals space 190 space cross times space 10 to the power of 9 space Pa
Diameter space of space the space iron space wire space equals space straight d subscript 1 space end subscript
Young ’ straight s space modulus space for space copper comma space straight Y subscript 2 space equals space 120 space cross times space 10 to the power of 9 space Pa space
Diameter space of space the space copper space wire space equals space straight d subscript 2

Therefore comma space

Ratio space of space their space diameters space is space given space by comma space

straight d subscript 1 over straight d subscript 2 space equals space square root of straight Y subscript 1 over straight Y subscript 2 end root space equals space square root of fraction numerator 190 cross times 10 to the power of 9 over denominator 120 cross times 10 to the power of 9 end fraction end root space equals square root of 19 over 12 end root