This is a seemingly basic mechanics problem but I'm having a dilemma in understanding what happens. You start with two blocks, one on top of the other (the bottom block has a larger mass than the top block). There is friction between the blocks, so they stick together. These two blocks rest on a frictionless surface. If I apply a force F to the top block, what happens?
Drawing the free body diagram, the force F I apply is counter acted by a frictional force f because of the friction between the two blocks. Additionally, this frictional force will be opposed by another frictional force on the bottom block that forces it in the direction I applied the force. Thus, we see that the blocks will move in the direction I have pushed.
HOWEVER, I can't understand why the top block will move forward as well. The force I have applied should be negated by the frictional force, but this is not the case. Clearly, the block accelerates forward despite the friction. Why is this paradox created???
Best Answer
I'm guessing your FBD looked something like this, where $F_1$ is the external force you apply. I'm assuming here that the top and bottom block don't slide relative to each other, so the forces at their junction ($F_2$) are equal and opposite.
The net force on the top block is the force you apply, $F_1$, minus the frictional force the bottom block applies to the top block, $F_2$:
$$ F_{top} = F_1 - F_2 $$
Because $F_1 > F_2$ the net force $F_{top} > 0$ and the top block accelerates.
Response to comment:
If the two blocks don't slide relative to each other then their accelerations must be the same so:
$$ \frac{F_{top}}{m_{top}} = \frac{F_{bottom}}{m_{bottom}} $$
We know that $F_{top} = F_1 - F_2$ and $F_{bottom} = F_2$, so:
$$ \frac{F_1 - F_2}{m_{top}} = \frac{F_2}{m_{bottom}} $$
and a quick rearrangement gives:
$$ F_1 = F_2 \frac{m_{top} + m_{bottom}}{m_{bottom}} $$
and since $m_{top} + m_{bottom} > m_{bottom}$ this means $F_1 > F_2$.
Basically $F_2$ is only accelerating the bottom block while $F_1$ is accelerating both blocks, so $F_1$ has to be greater than $F_2$.