A pulley fixed to the ceiling carries a thread with bodies of masses `m_1` and `m_2` attached to its ends. The masses of the pulley and the thread are negligible, friction is absent. Find the acceleration `w_C` of the centre of inertia of this system.

Explanation

We know that acceleration of centre of mass of the system is given by the expression.
`vecw_C=(m_1vecw_1+m_2vecm_2)/(m_1+m_2)`
Since `vecw_1=-vecw_2`
`vecw_C=((m_1-m_2)vecw_1)/(m_1+m_2)` (1)
Now from Newton's second law `vecF=mvecw`, for the bodies `m_1` and `m_2` respectively.
`vecT+m_1vecg=m_1vecw_1` (2)
and `vecT+m_2vecg=m_2vecw_2=-m_2vecw_1` (3)
Solving (2) and (3)
`vecw_1=((m_1-m_2)vecg)/(m_1+m_2)` (4)
Thus from (1), (2) and (4),
`vecw_c=((m_1-m_2)^2vecg)/((m_1+m_2)^2)`