Two Bodies in Contact on a Horizontal Surface

Consider two blocks of masses $m_1$ and $m_2$ ($m_1>m_2$) placed in direct contact on a smooth (frictionless) horizontal surface, and pushed together by a single external horizontal force $F$ applied to the outer face of one block. Since they stay in contact and move together, both blocks share the same acceleration $a$.

Treating both blocks together as one combined system of mass $m=m_1+m_2$:

To find the contact force between the two blocks (the force each exerts on the other via direct pushing contact — not a string tension), isolate just one block at a time. For the block not directly touched by $F$, Newton's second law gives: contact force $=$ (that block's own mass) $\times a$. So, for example, the force block 1 exerts on block 2 works out to

if $m_2$ is the far block (and symmetrically with $m_1$ and $m_2$ swapped if $m_1$ is the far block).

By Newton's third law, the contact force each block exerts on the other is equal and opposite: $\vec f_{12}=-\vec f_{21}$. …