In the arrangement shown in figure 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.

Explanation

Let us write Newton's second law for masses `m_1` and `m_2` and moving pully in vertical direction along positive x-axis (figure):
`m_1g-T=m_1w_(1x)` (1)
`m_2g-T=m_2w_(2x)` (2)
`T_1-2T=0` (as `m=0`)
or `T_1=2T` (3)
Again using Newton's second law in projection form for mass `m_0` along positive `x_1` direction (figure), we get
`T_1=m_0w_0` (4)
The kinematical relationship between the accelerations of masses given in terms of projection on the x-axis
`w_(1x)+w_(2x)=2w_0` (5)
Simultaneous solution of the obtained five equations yields:
`w_1=([4m_1m_2+m_0(m_1-m_2)]g)/(4m_1m_2+m_0(m_1+m_2))`
In vector form
`vecw_1=([4m_1m_2+m_0(m_1-m_2)]vecg)/(4m_1m_2+m_0(m_1+m_2))`