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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 ?
The maximum value of $Y$ that can be determined is $10^{14}\,N/m^2$
$\frac{{\Delta Y}}{Y}$ gets minimum contribution from the uncertainty in the length
$\frac{{\Delta Y}}{Y}$ gets its maximum contribution from the uncertainty in strain
The figure of merit is the largest for the length of the rod.
Solution
$Young's\,modulus\,Y = \frac{F}{A}/\frac{{\Delta \ell }}{\ell }$
$Y = \frac{{Fl}}{{\pi {r^2}\Delta l}}$
$Given,\,radius = 5mm,forceF = 50\pi KN,$
$\frac{\ell }{{\Delta \ell }} = 0.01\,mm$
$\therefore \,Y = \frac{F}{{\pi {r^2}}}\frac{\ell }{{\Delta \ell }} = 2 \times {10^{14}}N/{m^2}.$
