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Question
A wire elongates by 1.0 mm when a load W is hung from it. If this wire goes over a a pulley and two weights W each are hung at the two ends, he elongation of he wire will be
Options
0.5 m
1.0 mm
2.0 mm
4.0 mm
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Solution
\[\text{ Let the Young's modulus of the material of the wire be Y } . \]
\[\text{ Force = Weight = W (given) }\]
\[\text{ Let C . S . A . = A} \]
\[\text{ x = 1 mm = Elongation in the first case }\]
\[\text{ Length = L } \]
\[Y = \frac{\frac{W}{A}}{\frac{x}{L}} = \frac{WL}{Ax}\]
\[\text{Let y be the elongation on one side of the wire when put in a pulley }. \]
\[\text{ When put in a pulley, the length of the wire on each side }= \frac{L}{2}\]
\[ \frac{\frac{W}{A}}{\frac{y}{\frac{L}{2}}} = Y\]
\[ \Rightarrow \frac{\frac{W}{A}}{\frac{y}{\frac{L}{2}}} = \frac{WL}{\text{ Ax }}\]
\[ \Rightarrow y = \frac{x}{2}\]
\[\text{ Total elongation in the wire = 2y }= 2\left( \frac{x}{2} \right) = x = 1\text{ mm} \]
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