# Two stacked blocks with the upper block pulled with force $F$

I had to solve a question regarding the maximum force that can be applied on the upper of two stacked blocks for the blocks to move together (there is friction between the two blocks). I managed to solve the question, but am a bit confused conceptually.

When we consider a block on the ground that is pulled with a force $$F$$ (not larger than the maximum static friction), we say that the static friction force equals $$F$$ and so the block does not move. In the case of the stacked blocks here, as I understand it, F on the upper block does not produce an equal static frictional force. It seems like the force of static friction between the blocks here adjusts according to the F and the masses of the two blocks to ensure that the blocks have the same acceleration. Therefore, until a maximum F (where static friction is max), the two blocks will always accelerate together as the friction between the blocks takes up values such that this happens. Below this F, there is no possibility for the blocks to slide with respect to each other.

Is this understanding correct? It confuses me to think of static friction adjusting in such a complex manner (according to masses of the blocks and F) to allow joint movement of the two blocks. It seems like friction is ensuring there is no relative movement, instead of us explaining no relative movement via friction (if that makes sense).

• Are we to assume that the lower block is on a friction-less surface? Jan 11 '21 at 0:40
• @R.W.Bird yes there is no friction between the floor and the lower block
– ani
Jan 11 '21 at 4:43

## 1 Answer

The definition of the static friction is a force that opposes relative motion between two surfaces.

Thus, if you pull the top block with some force $$F$$, the static friction force will oppose sliding (such that the top and bottom blocks do not slide with respect to each other). This implies they must have the same acceleration (otherwise they'd slip).

This is also why $$F$$ won't equal the friction force in magnitude -- if that were so, the top block would remain at rest, whereas the bottom block would accelerate. That, obviously, cannot occur, since we already established that the blocks won't slip.

It confuses me to think of static friction adjusting in such a complex manner (according to masses of the blocks and F) to allow joint movement of the two blocks.

Think of static friction as a force that opposes relative motion between the surfaces. All other conclusions in this problem naturally flow from that definition. Hope this helps.

• A good analogy is the normal force. If a block with $10\; \rm N$ weight is resting on a flat surface, the normal force will be $N=mg$, therefore: $N=10\; \rm N$. However, if I push down on the block with a force of $5\; \rm N$, then the normal force will be $15\; \rm N$. This is just an example where you must do some 'extra' work to try to find the force. Same case with the static friction -- it's not always $\mu N$. Jan 10 '21 at 20:50