If the work done is stretching a wire by 1 mm is 2 J, the work necessary for stretching another wire of same material but with double the radius of cross section and half the length by 1 mm is
(a) 16 J (b) 8 J (c) 1 / 16 J (d) 1 / 8 J
\[\frac {1}{2} F \cdot \Delta L = W \]
\[Y=\frac{F/A}{\Delta L/L} \]
\[Y=\frac{F \cdot L}{A \cdot \Delta L} \]
\[F= \frac{Y A \cdot \Delta L}{L}\]
\[\frac {1}{2} Y A \cdot \frac {\Delta L}{L}\cdot\Delta L =2J\]
\[\frac{1}{2} Y \cdot \pi r^2 \cdot \frac {\Delta L}{L} \cdot \Delta L = 2 J \tag{1}\]
\[\frac {1}{2} Y \cdot \pi (2r)^2 \cdot \frac{\Delta L}{L/2} \cdot \Delta
L = W_2 \tag{2} \]
\[\frac {1}{2} Y \cdot \pi r^2 \cdot \frac {\Delta L}{L} \cdot \Delta L
\cdot 8 = W_2 \]
\[ 2 \times 8 = W_2 \]
\[16J = W_2\]
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