Two stacked bodies on a smooth floor: does the top body make the lower one accelerate more slowly?

Question

Please consider the following setting: body A is placed on the floor (which has no friction). Body B is placed on body A. There is static friction between the two.

A horizontal force P is applied on body A, making it accelerate to the left. Thanks to the static friction, body B comes along with it in the exact same speed and acceleration.

In terms of horizontal forces in this situation, my understanding is:

Body A experiences force P pushing it to the left. It also experiences the static friction force Fs[from-B-to-A], pushing it to the right.

Body B experiences the force Fs[from-A-to-B], pushing it to the left. The two Fs forces are of course identical in magnitude thanks to Newton's 3rd Law.

My question is:

Would A accelerate faster if it didn't have B on its back?

Logically, since the both are locked together, and there's no floor friction, I don't see why B should affect the acceleration of A.

However, looking at the forces, it seems to me that Fs[from-B-to-A] should play a role in reducing the acceleration value of A.

Answer 1

Would A accelerate faster if it didn't have B on its back?

If $A$ is not moving relative to $B$ it does not matter what forces act between them be they frictional, chemical bonds etc, you are comparing the acceleration of a mass $m_{\rm A}$ when acted on by a force $P$ with the acceleration of a mass $m_{\rm A}+m_{\rm B}$ when acted on by a force $P$.

$P= m_{\rm A}\,a_{\rm A}$ compared with $P= (m_{\rm A}+m_{\rm B})\,a_{\rm AB} \Rightarrow (m_{\rm A}+m_{\rm B})>m_{\rm A}$ thus $a_{\rm AB}< a_{\rm A}$

Note also that if the system is block $A$ and block $B$ the frictional forces are internal forces and a Newton's third law pair with resultant zero.

Answer 2

I think you have presented all the information you need pretty clearly. You are just failing at observing the result. I can see two ways of concluding the answer:

Option 1: Look at the system as a whole: you have two bodies with a combined mass, the two Fs forces cancel each other (as they are equal and opposite), so you end up with P moving the combined mass of A and B. The system will obviously accelerate slower than if A was on its own and subjected to the same P force.

Option 2: Look at A only: A is subjected to P force and one Fs force only (from B to A). Being in opposite direction, resulting force will be (P - Fs), giving a resulting force smaller than P, hence providing A body with less acceleration that what P would have provided. Note, that Fs from A to B is not contemplated in this case, as this is a force that is applied to body B only, not A.

Both ways of looking at it draw the same conclusion: body A will accelerate slower when body B is on top of it if subjected to the same P force.