Question

A thin $1 \,m$ long rod has a radius of $5\, mm$. A force of $50 \,\pi kN$ is applied at one end to determine its Young’s modulus. Assume that the force is exactly known. If the least count in the measurement of all lengths is $0.01\, mm$, which of the following statements is false ?

• A. $\frac{\Delta \, Y}{Y}$ get minimum contribution from the uncertainty in the length.
• B. The figure of merit is the largest for the length of the rod.
• C. The maximum value of Y that can be determined is $10^{14} \, N/m^2$.
• D. $\frac{\Delta \, Y}{Y}$ gets its maximum contribution from the uncertainty in strain.

Answer 1

Answer: (C)

$\ell=1 m$
$r=5 \times 10^{-3}\, m$
$F =50 \pi \times 10^{3}\, N$
$\gamma=\frac{ F / A }{\frac{\Delta \ell}{\ell}}$
$\gamma \,\frac{\Delta \ell}{\ell}=\frac{ F }{ A }$
$\gamma=\frac{50 \pi \times 10^{3}}{\pi \times\left(5 \times 10^{-3}\right)^{2}} \times \frac{\ell}{\Delta \ell}$
$\gamma=\frac{50 \times 10^{3}}{25 \times 10^{-6}} \times \frac{1}{\Delta \ell} $
$\Rightarrow \gamma=\frac{2 \times 10^{9}}{\Delta \ell}$
$\gamma=\frac{2 \times 10^{9}}{\varepsilon}$
$\gamma_{\max }=2 \times 10^{9}$