In the arrangement shown in Fig. 1.16 the bodies have masses $m_{0}, m_{1}, m_{2}$ , the friction is absent, the masses of the pulleys and the threads are negligible. Find the acceleration of the body $m_{1}$ . Look into possible cases.

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Solution

Let us write Newton's second law for masses $m_{1}$ and $m_{2}$ and moving pully in vertical direction along positive $x -$ axis (Fig.):
$m_{1} g - T = m_{1} w_{1 x}$
$m_{2} g - T = m_{2} w_{2 x}$
$T_{1} - 2 T = 0$ (as $m = 0$ )
or $T_{1} = 2 T$
Again using Newton's second law in projection from for mass $m_{0}$ along positive $x_{1}$ direction (Fig.), we get
$T_{1} = m_{0} w_{0}$
The kinematical relationship between the accelerations of masses give in term of projection on the $x -$ axis
$w_{1 x} + w_{2 x} = 2 w_{0}$
Simultaneous solution of the obtained give equations yields:
$w_{1} = \frac{4 m_{1} m_{2} + m_{0} (m_{1} - m_{2})] g}{4 m_{1} m_{2} + m_{0} (m_{1} + m_{2})}$
In vector form
$_{1} = \frac{[4 m_{1} m_{2} + m_{0} (m_{1} - m_{2})] \to g}{4 m_{1} m_{2} + m_{0} (m_{1} + m_{2})}$

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In the arrangement shown in Fig. 1.16 the bodies have masses m0,m1,m2, the friction is absent, the masses of the pulleys and the threads are negligible. Find the acceleration of the body m1. Look into possible cases.

In the arrangement shown in Fig. 1.16 the bodies have masses $m_{0}, m_{1}, m_{2}$ , the friction is absent, the masses of the pulleys and the threads are negligible. Find the acceleration of the body $m_{1}$ . Look into possible cases.