Find the acceleration of the body of mass $m_2$ in the arrangement shown in figure. If the mass $m_2$ is $\eta$ times great as the mass$m_1$, and the angle that the inclined  plane forms with the horizontal is equal to $\theta$. The masses of the pulleys and threads, as well as the friction is negligible.

 Here, by constraint relation we can see that the acceleration of $m_2$ is two times that of $m_1$. So, we assume if $m_1$is moving up the inclined plane with an acceleration a, the acceleration of mass $m_2$ going down is $2a$. The tensions in different strings are shown in figure.

The dynamic equations can be written as

For mass $m_1$ :

$2T-m_{1}g\sin \theta =m_{1}a$                                   ………………………..(i)

For mass $m_2$ :

$m_{2}g-T=m_2\left(2a\right)$                                            ……………………(ii)

Substituting $m_2=\eta m_1$ and solving Eqs. (i) and (ii), we get

Acceleration of $m_2=2a=\frac{2g(2\eta -\sin \theta )}{4\eta +1}$