A thin 1 m long rod has a radius of 5 mm. A force of 50 $\mathrm{\pi }$kN is applied at one end of 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? 

Given,

$\mathrm{F}=50\mathrm{\pi kN}, \mathrm{r}=5\mathrm{mm}, \mathrm{L}=1\mathrm{m}$

we know that Young's modulus,

$\mathrm{Y}=\frac{\mathrm{FL}}{\mathrm{A}l}$

substituting the values,

$\mathrm{Y}=\frac{50\mathrm{\pi }\times {10}^3\times 1}{\mathrm{\pi }\times 25\times {10}^{-6}\times l}=\frac{2\times {10}^9}{l}$

The Young's modulus will have a maximum value of $2\times {10}^9.$
The change in young's modulus ratio will then be calculated as $\frac{∆\mathrm{Y}}{\mathrm{Y}}=2\frac{∆\mathrm{r}}{\mathrm{r}}+\frac{∆l}{l}$.

If we look at this equation, we can see that the uncertainty in length contributes the least to changes in young's modulus and the uncertainty in radius contributes the most to changes in young's modulus.

Hence the correct answer is The maximum value of $\mathrm{\gamma }$ that can be determined is 10¹⁴N/m².