Two blocks of mass $2\ \text{kg}$ and $4\ \text{kg}$ hang from the two ends of a metal wire going over a smooth fixed pulley. The radius of the wire is $4.0\times10^{-5}\ \text{m}$ and Young's modulus of the metal is $2.0\times10^{11}\ \text{N m}^{-2}$. The longitudinal strain developed in the wire is $\dfrac{1}{\alpha\pi}$. The value of $\alpha$ is ______. [Use $g = 10\ \text{m s}^{-2}$]
Numerical value type. Enter your answer.
Answer: 12
Tension in an Atwood machine:
$$T = \frac{2m_1m_2g}{m_1 + m_2} = \frac{2\times2\times4\times10}{6} = \frac{80}{3}\ \text{N}$$
$$\text{strain} = \frac{T}{\pi r^2Y} = \frac{80/3}{\pi\times16\times10^{-10}\times2\times10^{11}} = \frac{80/3}{320\pi} = \frac{1}{12\pi}$$
$\alpha = 12$.
Solution by Sreeraj P, M.Sc Physics