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Byju's Answer

Standard XII

Physics

Elastic Potential Energy

In the arrang...

Question

In the arrangement shown in the figure weight $A$ possesses mass $m$ , a pulley $B$ possesses mass $M$ . Also known are the moment of inertia $I$ of the pulley to its axis and the radii of the pulley are $R$ and $2 R$ respectively. Consider the mass of the threads is negligible. Find the acceleration of weight $A$ after the system is set free. (Assume no slipping takes place anywhere and axis of cylinder remains horizontal).

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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 the figure. For the cylinder from the equation $F_{x} = m w$ and $N_{e x} = I_{c} β_{z}^{'}$ we get
$M g + T_{A} - 2 T = M w_{c}$
and $2 T R + T_{A} (2 R) = I β = \frac{I w_{c}}{R}$
For the weight $A$ from the equation
$F_{x} = m w_{x}$
$m g - T_{A} = m w_{A}$
As there is no slipping of the threads on the pulleys.
$w_{A} + w_{c} + 2 β R = w_{c} + 2 w_{c} = 3 w_{c}$
Simultaneous solutions of above equations gives:
$w_{A} = \frac{3 (M + 3 m) g}{(M + 9 m + \frac{I}{R^{2}})}$

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