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Com in Perfectly Inelastic Collision

In the arrang...

Question

In the arrangement shown in figure above, a weight $A$ possesses mass $m$ , a pulley $B$ possesses mass $M$ . Also known are the moment of inertia $I$ of the pulley relative to its axis and the radii of the pulley $R$ and $2 R$ . The mass of the threads is negligible. If the acceleration of the weight $A$ after the system is set free is $w = \frac{b (M + b m) g}{(M + 9 m + \frac{I}{R^{2}})}$ , find the value of $b$ .

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Solution

Let us depict the forces acting on the pulley and weight $A$ , and indicate positive direction for $x$ and $φ$ as shown in figure below. For the cylinder from the equation $F_{x} = m w_{c x}$ and $N_{c z} = I_{c} β_{z}$ , we get
$M g + T_{A} - 2 T = M w_{c}$ ....(1)
and $2 T R + T_{A} (2 R) = I β = \frac{I w_{c}}{R}$ .....(2)
For the weight $A$ from the equation
$F_{x} = m w_{x}$
$m g - T_{A} = m w_{A}$ ......(3)
As there is no slipping of the threads on the pulleys,
$w_{A} = w_{c} + 2 β R = w_{c} + 2 w_{c} = 3 w_{c}$ .....(4)
Simultaneous solutions of above four equations gives,
$w_{A} = \frac{3 (M + 3 m) g}{(M + 9 m + \frac{I}{R^{2}})}$

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