# Why doesn't the upper block move when force less than limiting friction is applied, in two block problem (further explanation below)?

In my school, I learned that when two blocks are placed on the ground with one block above the other, if a force is applied to the lower block, two opposing forces of friction act on it: one from the ground and the other from the upper block's surface. Consequently, according to Newton's third law, the upper block experiences a friction force in the forward direction. However, I have a question regarding this scenario. If the external force applied to the lower block is significantly less than the limiting friction of the ground, the lower block won't be set into motion due to the opposition from the static friction of the ground. In addition, I believe that the static friction of the upper block also plays a role in opposing the motion(as it does when the block do move). Consequently, the upper block should experience an equal and opposite reaction that sets it into motion as well. However, this doesn't seem to happen in reality. What misconception do I have in this situation?

• As my old mechanics professor used to emphatically say, "Draw a free body diagram!!." Commented Jun 12, 2023 at 11:22
• I will be submitting an answer shortly. Commented Jun 12, 2023 at 21:31

In my school, I learned that when two blocks are placed on the ground with one block above the other, if a force is applied to the lower block, two opposing forces of friction act on it: one from the ground and the other from the upper block's surface.

It is correct that the friction force from the ground opposes the applied force on the lower block (block b), but there will be no friction forces between the two blocks unless the lower block is set into motion. If the ground friction force is static friction, it will match the applied force $$F$$ for a net horizontal force of zero and there will be no force causing friction to arise between the blocks. Only if the maximum possible static friction force between the lower block and the ground is exceeded setting the lower block into motion, will there be friction between the blocks to oppose such motion. It is important to understand that static friction only exists in opposition to a net force that would act on the object in the absence of any friction.

Consequently, according to Newton's third law, the upper block experiences a friction force in the forward direction.

The upper block will experience a friction force in the forward direction only if the lower block experiences a friction force in the backwards direction by the upper block. But as indicated above, there will be no backwards direction friction force on the lower block unless the maximum possible static friction force on the ground is exceeded so that the lower block accelerates forward.

However, I have a question regarding this scenario. If the external force applied to the lower block is significantly less than the limiting friction of the ground, the lower block won't be set into motion due to the opposition from the static friction of the ground.

That is correct.

In addition, I believe that the static friction of the upper block also plays a role in opposing the motion(as it does when the block do move).

That is correct, but only in opposing motion of the lower block that is actually occurring (sliding), not in preventing motion of the lower block from occurring. When the lower block is set into motion, the ground friction becomes kinetic, which is generally less than the static friction that initiated the motion, and the applied force $$F$$ on the lower block will be greater than the kinetic friction force, call it $$f_{kG}$$, from the ground on the lower block. Now there will be a net force on the lower block for static friction imposed by the upper bloc to the lower block, call it $$f_{s-ab}$$, to oppose. Then the net force on the lower block becomes:

$$F-f_{kG}-f_{s-ab}$$

And from Newton's 2nd law its acceleration becomes:

$$a_{b}=\frac{(F-f_{kG}-f_{s-ab})}{m_a}$$

Consequently, the upper block should experience an equal and opposite reaction that sets it into motion as well. However, this doesn't seem to happen in reality. What misconception do I have in this situation?

The misconception originates from the original statement of what you say you learned in school. There is no static friction force acting on the lower block by the upper block's surface unless the lower block is set into motion, as I discussed above.

In closing I suggest in the future your first step should be to draw free body diagrams for the individual blocks. In this example, one where the system is stationary (i.e., the maximum possible static friction force between the lower block and ground is not exceeded). The other for when the lower block is set into motion (i.e., the maximum ground static friction is reached and friction becomes kinetic) and you want to determine whether the two block move together with the same acceleration, or the top blocks slides on the lower when the applied force us great enough, as in the example of the "table cloth trick" shown in this video: https://www.google.com/search?q=table+cloth+trick&oq=table+cloth+trick&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQABiABDIJCAIQABgKGIAEMgkIAxAAGAoYgAQyCQgEEAAYChiABDIJCAUQABgKGIAEMgkIBhAAGAoYgAQyCQgHEAAYChiABDIJCAgQABgKGIAEMgoICRAAGAoYFhge0gENMjA2OTAwMjFqMGoxNagCALACAA&sourceid=chrome&ie=UTF-8

Hope this helps.

In addition, I believe that the static friction of the upper block also plays a role in opposing the motion(as it does when the block do move).

The friction of the upper block is only opposing the motion between the blocks. i.e. You are missing Galilean relativity.

There will not be any friction between the blocks if the two blocks are still stationary, i.e. if you are just starting to push the lower block, and the friction from the floor is enough to stop the blocks, then there will be no friction between the blocks.

In this scenario, really, what the friction between the blocks do, is to enforce the constraint that the two blocks move as one. It just changes the acceleration of the two blocks to make them equal.

• When will I know that static friction and Kinetic friction acts between the ground and lower block and when static friction and Kinetic acts between the blocks Commented Jun 12, 2023 at 6:50
• It might be helpful to the OP if you could explain why block b can't slide under block a, i.e., why the two blocks must move as one. Commented Jun 12, 2023 at 13:01

This will supplement my answer regarding your initial statement by providing a simple example.

Imagine a chair on a floor with a book on the chair. The chair and book are simply sitting there. No horizontal forces are being applied to either. Although the contact surfaces are rough (at least microscopically) no friction forces exist parallel to the surfaces because there are no applied forces for friction to resist.

Now you apply a slight force on the chair parallel to the floor. The chair doesn’t move. Why? Because there is now a static friction force acting on the chair by the ground equal and opposite to your force.

What about the book? Does it move? No. In order for it to move forward an external force would need to act forward on it. A free body diagram of the book would show that the only potential external horizontal force acting forward on the book would be a friction force exerted by the surface of the chair. The fact that the book does not move means there’s no friction force acting on it by the chair. And if that’s true it means, by Newton’s 3rd law, there is no equal and opposite friction force acting backwards on the chair.

In conclusion, the only forces acting on the chair if it doesn’t move in response to your force, is the applied force and the equal and opposite ground friction force.

Hope this helps.

• Hey pls answer to my latest question as well! Commented Jun 16, 2023 at 3:27
• @AyushNaman Your latest question is closed. Answers can no longer be given. Commented Jun 16, 2023 at 12:56
• could you please provide the answer in short over here. I would really appreciate your help. Commented Jun 16, 2023 at 15:30
• @AyushNaman You're asking me to evade the policy of this site. Commented Jun 16, 2023 at 17:50
• "No worries, thanks anyways." Commented Jun 16, 2023 at 18:45