Condition for stacked bodies to *move together* under friction

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I was given a question in a mock test series:

Three blocks A, B and C of masses $2\,\text{kg}$, $3\,\text{kg}$ and $4\,\text{kg}$ are placed as shown. Coefficient of friction between A and B is 0.5 and that between B and C is 0.1. Maximum force $F$ that can be applied horizontally on to A such that three blocks move together is _________ ($g=10\,\text{m}/\,\text{s}^2$).

Options:

• $12.22\,\text N$
• $13\,\text N$
• $11.25\,\text N$
• $15\,\text N$

My beef is the part where they ask us to find to force so that the blocks move together. Does this make any sense?

If we assume that by move together, it is meant that the blocks remain in same relative position to each other, the calculations come out wrong.
From the options we know that, $F>10\,\text{N}$. As such, if move together means block remain in same relative position to each other, then the acceleration of all the blocks must be same, i.e $$\frac{F-f_{ab}}{m_A}=\frac{f_{ab}-f_{bc}}{m_B}=\frac{f_{bc}}{m_C}$$

We know, $F>10\,\text{N}$ and $f_{{ab}_{\text{max}}}=10\,\text{N}$. So $f_{ab}=f_{{ab}_{\text{max}}}=10\,\text{N}$. Again $f_{ab}>f_{{bc}_{\text{max}}}=5\,\text N$. So $f_{bc}=5\,\text N$ Hence now we know the values.

But on placing the values, the condition is not fulfilled. Hence I don't think this is what the question means.

Hence what does move together mean in this context? Can someone help me out?

Answer 1

Your mistake is assuming that the frictional force $f_{AB}$ tries to fully counter the push $F,$ such that it reaches its maximum ($|f_{AB}|\leq\mu_{AB}m_Ag=10\,\mathrm{N}$) whenever $F>10\,\mathrm{N}.$ It doesn't; even when $F<10\,\mathrm{N},$ $f_{AB}$ does not completely counter it. That just isn't a valid physical principle. Rather, you should start with your relation $$\frac{F-f_{AB}}{m_A}=\frac{f_{AB}-f_{BC}}{m_B}=\frac{f_{BC}}{m_C}.$$

This correctly describes a necessary and sufficient condition for "the blocks moving together." You can solve this equation for $f_{AB},f_{BC}$ to find the frictional forces that would be necessary for the blocks to move together (as a function of $F$): $$\begin{gathered}f_{AB}=\ldots\text{(something involving $F$)}\ldots\\f_{BC}=\ldots\text{(something involving $F$)}\ldots\end{gathered}$$ Then you may take the inequalities $f_{AB}\leq f_{AB,\text{max}},f_{BC}\leq f_{BC,\text{max}}$ and turn them into bounds on $F$. You find one of the answer choices.