Confusion between pressure and force

Question

If I had two pistons of areas $A_1$ and $A_2$ where $A_1 > A_2$, and I pushed them with the same force against one another, would they be in an equillibrium and cancel each other out, or will the piston with area $A_2$ win because it has greater pressure?

The reason why I'm asking is because when we talk about fluids like air or water, we always talk in terms of pressure, and when there's a higher pressure on one side and lower pressure on the other side, fluids go to the lower pressure, but why would that be the case if the forces are the same and the areas are different?

Also, take a nail piercing through a wall. If I applied the same force on a nail, why would surface area matter? If the force applied is the same, how can the wall crack just by deacreasing the surface area of the applied force, if force is defined as a thing that makes the objects it is exerted on accelerate?

Answer 1

Pressure is defined as the normal force acting on the surface divided by the area on which it acts. Stress is a more general term which does not necessarily need normal force ( the force can be parallel or at some angle to the area).

1. The motion is caused by force and that force can be due to the pressure difference. When you have connected two cylinders of different area and apply same force ( as if trying to compress them together), there will be equilibrium. However, that also means that the cylinders will experience pressure (compressive stress to be more precise) acting on them. Since the area of contact between the cylinders is same (area of the smaller cylinder A2 in this case), stress would be same. When you introduce the third cylinder with transitioning diameter in between, still the system of three cylinders will be in equilibrium but stresses ( or pressure if you say) will vary in that transitioning cylinder. ( it will be constant in the two pistons)

2. In case of a fluid, assume a pipe with a constant diameter. If you apply pressure more on the right side than the left (it is the pressure difference that matters), you need to multiply that difference in pressure with the cross-sectional area of the tube to get the force. Note that this force is the Net force i.e. the difference of forces due to pressure on the right side and left side acting on the fluid. The force is NOT the same throughout the pipe and that's why fluid goes from high pressure to lower pressure.

3. In the case of piercing the nail, your objective is to initiate a crack and make way for the nail to go inside the material. Every material has certain mechanical properties ( e.g. Hardness: defining the resistance a material poses for indentation i.e. piercing). Hardness can also be visualized as resistance to the deformation of the local region. A crude way to say is you need enough stress (https://en.wikipedia.org/wiki/Stress%E2%80%93strain_curve )so that material is able to deform locally and make way for the nail to go ( things get more complicated when plasticity gets into the way). Hence having a smaller area helps you achieve more stress at the tip and hence exceed the critical stress value to deform the material locally.

Answer 2

If two equal forces are opposed, they are in equilibrium. The side with greater pressure does not win.

Imagine fluids at different pressures on two sides of a piston. If the faces of the piston have the same area then the side with the higher pressure applies the higher force. So the piston moves in the direction in which the higher force is pointing, which is also the direction from higher to lower pressure.

If the faces of the piston have different areas but the same pressure, the side with the bigger area wins, because it has the bigger force.

If the faces of the piston have different areas and different pressures, you have to multiply them to find out which side has the bigger force. The side with the higher force always wins.

Air moves from high to low pressure because if two masses of air are in contact then they have the same contact area, so the higher pressure is pushing with the higher force.

A similar thing happens with the nail. One end (the head) is broad, the other end is pointed. This is like the piston with a large face on one side and a narrow face on the other. If the same pressure is applied to both faces, the broad face wins because it has the larger area so it has the larger force. If a force is applied at the broad end, then in order to oppose it the same force has to be applied at the pointed end. But there is very little material in contact with the pointed end, so all of the force from the broad end is concentrated at the point : the pressure is much higher at the point than at the head. If the material is not strong enough it will break or split and be pushed aside by the point of the nail.