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Suppose I have a block of mass $m$ on top of a block of mass $M $. Friction exists between them. I apply a force $F$ on the top block. The equations of motion are thus $$F-f_{frictional}=ma$$
$$f_{frictional}=Ma $$Consider the case when $F=f_{frictional} $. Then acceleration of the top block is 0, however the bottom block has an acceleration $a=\frac {F} {M} $. Doesn't it seem unlikely? What am I missing? P.S. No friction between bottom block and surface.
If $F=f_{friction}$ then, as you say, we must have $a=0$ and so from $f_{friction}=Ma$ we can conclude that $F=f_{friction}=0$ i.e. there is no force acting on the upper block.
If, however, $F>0$ then we must have $F>f_{friction}$, and we need to find values of $f_{friction}$ and $a$ that satisfy the simultaneous equations
$F-f_{friction} = ma \\ f_{friction}=Ma$
These equations are both satisfied when
$\displaystyle a = \frac {F}{M+m} \\ \displaystyle f_{friction} = F \frac {M}{M+m}$
So, as expected, we have $a>0$ and
$\displaystyle F = f_{friction} \frac{M+m}{M} = f_{friction} \left( 1+ \frac{m}{M} \right) > f_{friction}$
Consider the case when $F=f_{frictional} $. Then acceleration of the top block is 0, however the bottom block has an acceleration $a=\frac {F} {M} $. Doesn't it seem unlikely? What am I missing?
What you are missing is the applied force $F$ cannot equal the friction force. When approaching problems like this it is essential to draw a free body diagram (FBD). See the FBD below. It is based on your subsequent clarification that there is no friction between the bottom block and the surface underneath.
As long as the maximum possible static friction force, $\mu_{s}mg$, between the blocks is not exceeded the two blocks will move together. Ignoring all the vertical forces which sum to zero, there are two forces acting on the top block, the applied external force $F$ and the static friction force $f_f$ that the lower block applies to the top. There is only one external force acting on the bottom block, the friction force applied to it by the top block which is equal and opposite to the friction force applied to the top block by the bottom block.
The acceleration of the top block is
$$a_{m}=\frac {F-f_{f}}{m}$$
The acceleration of the bottom block is
$$a_{M}=\frac {f_f}{M}$$
As long as $F$ doesn't exceed the maximum static friction force between the blocks, the top and bottom blocks move together and therefore
$$a_{m}=a_{M}$$
Equating the two accelerations and doing some math gives us
$$f_{f}=\frac{F}{(1+m/M)}$$
The two forces $F$ and $f_f$ are therefore not equal.
Hope this helps.
When $F=f_\mathrm{friction}$ or $F<f_\mathrm{friction}$, the friction cannot be overcome, i.e., the top block will not move (no acceleration from rest). But, the applied force $F$ will be "transferred" into the bottom block, which then experiences an acceleration $a=F/(M+m)$, since both masses act as one in this case. As an example, consider sitting on a bobsled with a "grippy" rubber layer on top, which in turn rests on an icy surface (so the friction between the ground and the sled is basically 0). If your friend pushes you, he will also move the sled underneath you by transferring the force onto both the sled and you. But in absence of a grippy rubber layer and instead a slippery bare metal surface, he would just push you off the sled (of course, there is still some friction in real life, consider it just for illustration purposes).
