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In the arrang...

Question

In the arrangement shown in the figure the mass of the uniform solid cylinder of Radius $R$ is equal to $m$ and the masses of two bodies are equal to $m_{1}$ and $m_{2}$ . The thread slipping and the friction in the axel of the cylinder are supposed to be absent. Find the angular acceleration of the cylinder and the ratio of tensions $\frac{T_{1}}{T_{2}}$ of the verticle sections at the thread in the process of motion.

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Solution

In the system above,
$a_{1} = a_{2}$ [By constraint]
Also, $a_{1} = a_{2} = α r$ [No slipping]
Equation on thread be,
$m_{2} g - T_{2} = m_{2} a_{2} ⟶ (1), m_{1} g - T_{1} = m_{1} a_{2} ⟶ (1)$
Also from torque equation, due to tension over cylinder,
$T_{2} R = T_{1} R = \frac{M R^{2}}{2} \times α \to (3) [τ = i α]$
From $(1), (2)$ and $(3)$ ,
$α = \frac{(m_{2} - m_{1}) g}{(m_{1} + m_{2} + \frac{m}{2}) g}$
and, $\frac{T_{1}}{T_{2}} = \frac{m_{1} g + m_{1} g_{2}}{- m_{2} g_{2} + m_{2} g} = \frac{m_{1} (m + 4 m_{2})}{m_{2} (m + {4 m}_{1})}$ .

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