Two blocks $m_{1}$ and $m_{2}$ are acted upon by the forces $F_{1}$ and $f_{2}$ as shown in Fig. If there is no relative sliding between the blocks and the ground is smooth, find the static friction between the blocks. Make the necessary assumptions and discuss different cases.

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Solution

As there is no sliding between the blocks, the friction will be static. Assuming the frictional forces on $m_{1}$ as $f (\leftarrow)$ and $m_{2}$ as $f (\to)$ , we have drawn the FBD without normal reactions and weights.
Referring to FBD as shown in Fig.
We have the following equations of motion:
For $m_{1} : F_{1} - f = M_{1} a_{1}$ (i)
For $m_{2} : F_{2} = m_{2} a_{2}$ (ii)
Since the friction is static, $a_{1} = a_{2}$
Substituting $a_{1}$ from Eq. (i) $a_{2}$ from Eq. (ii) in (iii), we have
$\frac{F_{1} - f}{m_{1}} = \frac{F_{2} - f}{m_{2}}$
This gives $f = \frac{m_{2} F_{1} - m_{1} F_{2}}{m_{1} + m_{2}}$

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