8) A light inextensible string is connected at its ends to two particles of masses \( m_{1} \) and \( m_{2}\left(m_{1}>m_{2}\right) \) and passes over a uniform circular pulley of radius \( a \) which can rotate freely about a fixed horizontal axis through its centre. The particles hang freely and the system is released from rest. If the pulley is sufficiently rough to prevent the string slipping, find the acceleration of either particle. (For the pulley \( I=\frac{1}{2} M a^{2} \).) The heavy pulley is now replaced by a light smooth one and both particles have their masses increased by the same amount, \( m \). If the acceleration of the particles is the same as it was in the first case, find an expression for \( m \).
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An object of mass $m_{1}$ hangs from a string that passes over a very light fixed pulley $\mathrm{P}_{1}$ as shown in Figure P5.34. The string connects to a second very light pulley $\mathrm{P}_{2} .$ A second string passes around this pulley with one end attached to a wall and the other to an object of mass $m_{2}$ on a frictionless, horizontal table. (a) If $a_{1}$ and $a_{2}$ are the accelerations of $m_{1}$ and $m_{2},$ respectively, what is the relation between these accelerations? Find expressions for (b) the tensions of in the strings and $(c)$ the accelerations $a_{1}$ and $a_{2}$ in terms of the masses $m_{1}$ and $m_{2},$ and $g .$
An object of mass $m_{1}$ hangs from a string that passes over a very light fixed pulley $P_{1}$ as shown in Figure $P 5.34$ The string connects to a second very light pulley $P_{2}$. A second string passes around this pulley with one end attached to a wall and the other to an object of mass $m_{2}$ on a frictionless, horizontal table. (a) If $a_{1}$ and $a_{2}$ are the accelerations of $m_{1}$ and $m_{2},$ respectively, what is the relation between these accelerations? Find expressions for (b) the tensions in the strings and (c) the accelerations $a_{1}$ and $a_{2}$ in terms of the masses $m_{1}$ and $m_{2},$ and $g$
Two masses $8 \mathrm{~kg}$ and $12 \mathrm{~kg}$ are connected at the two ends of a light inextensible string that goes over a frictionless pulley. The acceleration of the masaes and the tension in the string when the masses are released, are respectively [NCERT] (a) $2 \mathrm{~m} / \mathrm{s}^{2}$ and $90 \mathrm{~N}$ (b) $4 \mathrm{~m} / \mathrm{s}^{2}$ and $90 \mathrm{~N}$ (c) $2 \mathrm{~m} / \mathrm{s}^{2}$ and $60 \mathrm{~N}$ (d) $4 \mathrm{~m} / \mathrm{s}^{2}$ and $99 \mathrm{~N}$
Laws of Motion and Friction
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