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(D) For the solid cylinder of mass $m_2$,the forces acting are gravity $(m_2 g)$ downwards and tension $(T)$ upwards. The equation of motion for linea

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None of these

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(D) For the solid cylinder of mass $m_2$,the forces acting are gravity $(m_2 g)$ downwards and tension $(T)$ upwards. The equation of motion for linear acceleration $a_2$ is:$m_2 g - T = m_2 a_2$  --- $(1)$The torque equation about the center of the cylinder is:$T R = I \alpha_2 = (\frac{1}{2} m_2 R^2) \alpha_2$  --- $(2)$From equation $(2)$,we get:$T = \frac{1}{2} m_2 R \alpha_2$Substitute $T$ into equation $(1)$:$m_2 g - \frac{1}{2} m_2 R \alpha_2 = m_2 a_2$Divide by $m_2$:$g - \frac{1}{2} R \alpha_2 = a_2$Rearranging for $\alpha_2$:$\frac{1}{2} R \alpha_2 = g - a_2$$\alpha_2 = \frac{2(g - a_2)}{R}$Since the provided options do not match this result,the correct answer is None of these.

In the given figure,the linear acceleration of the solid cylinder of mass $m_2$ is $a_2$. If there is no slipping,find its angular acceleration $\alpha_2$.

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