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(B) Young's modulus $Y$ is given by $Y = \frac{FL}{A \Delta L} = \frac{MgL}{\pi (d/2)^2 \Delta L} = \frac{4MgL}{\pi d^2 \Delta L}$.Given: $M = 10 \, k

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$(2.0 \pm 0.2) 	imes 10^{11} \, N/m^2$

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(B) Young's modulus $Y$ is given by $Y = \frac{FL}{A \Delta L} = \frac{MgL}{\pi (d/2)^2 \Delta L} = \frac{4MgL}{\pi d^2 \Delta L}$.Given: $M = 10 \, kg$,$L = 2 \, m$,$g = 9.8 \, m/s^2$,$d = 0.4 	imes 10^{-3} \, m$,$\Delta d = 0.01 	imes 10^{-3} \, m$,$\Delta L = 0.88 	imes 10^{-3} \, m$,$\delta(\Delta L) = 0.05 	imes 10^{-3} \, m$.Calculating $Y$: $Y = \frac{4 	imes 10 	imes 9.8 	imes 2}{3.14 	imes (0.4 	imes 10^{-3})^2 	imes 0.88 	imes 10^{-3}} = \frac{784}{3.14 	imes 0.16 	imes 10^{-6} 	imes 0.88 	imes 10^{-3}} \approx 1.78 	imes 10^{11} \approx 2.0 	imes 10^{11} \, N/m^2$.Relative error: $\frac{\Delta Y}{Y} = 2 \frac{\Delta d}{d} + \frac{\Delta(\Delta L)}{\Delta L} = 2 \left( \frac{0.01}{0.4} ight) + \frac{0.05}{0.88} = 0.05 + 0.0568 \approx 0.1068$.Absolute error: $\Delta Y = 0.1068 	imes 2.0 	imes 10^{11} \approx 0.21 	imes 10^{11} \approx 0.2 	imes 10^{11} \, N/m^2$.Thus,$Y = (2.0 \pm 0.2) 	imes 10^{11} \, N/m^2$.

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