System of Masses $m_1 + m_2 + m_3 $

Three masses $m_1 , m_2 , m_3 $ kept on a smooth horizontal surface under the influence of force F have got certain acceleration (refer figure). Find the force that mass $m_1$ exerts on mass $m_2$.

Solution

Let a be the acceleration.

Taking $m_1 + m_2 + m_3 $ as system,

$F = (m_1 + m_2 + m_3)a$

$\Rightarrow a = \frac {F}{m_1 + m_2 + m_3 }$

Let N be the force that $m_1 $ exerts on $m_2 $. Taking $m_2 + m_3$ as system,

$N = (m_2 + m_3 ) a$

$\Rightarrow N = F. \frac {m_2 + m_3 }{m_1 + m_2 + m_3 } $

Interestingly, as $m_1 \to 0$, $N \to F$