Question

A thin $1m$ long rod has a radius of $5mm$. 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 $0.01mm$, 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)

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