Question

A thin 1 m long rod has a radius of 5 mm.1 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) the maximum value of Y that can be determined is ${{10}^{14}}N{{m}^{-2}}$
b) change in Y gets minimum contribution from uncertainty in length
c) change in Y gets maximum contribution from strain
d) the figure of merit is largest for length of rod

Answer 1

Hint: Let us first find the young’s modulus of the long rod with given quantities. Next, change in the young’s modulus is found out in relation with strain, length and radius etc. finally, we need to find what’s contribution is more and less towards the young’s modulus change. We can easily find out.

Formula used: \[Y=\dfrac{F}{A}\]

Complete step by step answer:
Let us first write the given terms,
$F=50\pi kN,r=5mm,L=1m$
Applying these terms in young’s modulus formula, we get,
$\begin{align}
  & Y=\dfrac{FL}{Al} \\
 & \Rightarrow Y=\dfrac{50\pi \times {{10}^{3}}\times 1}{\pi \times 25\times {{10}^{-6}}\times l} \\
 & \Rightarrow Y=\dfrac{2\times {{10}^{9}}}{l} \\
\end{align}$
If we observe the Young’s modulus, the maximum value will be $2\times {{10}^{9}}$
Next, the change in young’s modulus ratio will be taken as,
$\dfrac{\Delta Y}{Y}=2\dfrac{\Delta r}{r}+\dfrac{\Delta l}{l}$
If we observe this equation, we can find out that the minimum contribution to change in young’s modulus is given by the uncertainty in the length and the maximum contributing to the uncertainty in young’s modulus is given by uncertainty in radius.

So, the correct answer is “Option A”.

Additional Information: Young’s modulus or dur modulus of elasticity in tension, is a mechanical property that measures the tensile stiffness of solid material. It counts the relationship between tensile stress and axial strain in the linear elastic region of a material. Young’s modulus represents the factor of proportional teen Hooke’s law, which relates the stress and the strain. if very small stresses or strains are applied to a non-linear material, the response will be linear, but if very high stress or strain is applied to a linear material, the linear theory will not be enough.

Note: The young’s modulus depends on the force acting on the body and the area of the body. The uncertainty in the young’s modulus will depend on all the factors like radius, length of the material, area of the material etc. Also, in the above question, the uncertainty contribution is taken according to the percentage of the quantity for the total value.