# Using Newton's third law

A piston is free to move in a cylinder which contains a gas. The gas exerts a force of 10N on the piston, and thus by Newton's 3rd law, the piston exerts a force of 10N on the gas. Hence the gas and piston move in opposite directions, which is contradictory. How do I properly apply Newton's 3rd law to a case like this?

Also, this is completely different from what the intended question, but does the expression $$W = PV$$ apply to liquids or solids expanding? Does it apply when solids are not expanding, but are being moved, for e.g a cube is physically moved 4 meters by a constant horizontal force of 10N?

• Possible dupe physics.stackexchange.com/q/45653 Apr 23, 2017 at 18:30
• .. Read the question again. I know of the fallacy by which there is no motion, but that is not at all what I'm asking.
– John
Apr 23, 2017 at 20:21
• @sanjitSarda By the way, "metres" is a completely acceptable spelling. There's no need to edit it to "meters".
– JMac
Feb 3, 2020 at 14:27
• @JMac Thanks! I somehow never knew that. Feb 4, 2020 at 2:31

Hence the gas and piston move in opposite directions

No, this statement was too fast. Yes, they both feel the same force, but one force does not give motion. The sum of forces gives motion, according to Newton's 2nd law.

Since the piston is free to move, it moves, but the gas is not free to move since the container-wall behind it holds back. End result is that the net force on the gas is zero and it doesn't move (except for expanding) but the piston with non-zero net force does.

• Thanks for the answer! It makes sense now, but one thing I don't get it is whether expansion is because of a force or not.. I guess what I'm trying to ask is the reason gases fill up their container. Edit: Could you answer the second ( unrelated ) part as well?
– John
Apr 23, 2017 at 22:09
• Actually, I don't get it. If the molecules of the gas in contact with the piston exert a force on it, then the piston exerts a force on them in the opposite direction. They then move wherever this force points. So by this logic, shouldn't the gas contract?
– John
Apr 24, 2017 at 17:02
• @Saad That gas expands is due to an internal force - let's call it a "thermal force" - because energetic particles gain kinetic energy when e.g. heated and thus move more violently around, pushing harder in their surroundings. The fact that they are constantly moving (faster when hot, slower when cold, but nevertheless moving) also answers the question of why a gas always fills it's container; gas particles will simply naturally drift towards an area where they have less resistance to move around. That would be an empty space (a vacuum). Apr 24, 2017 at 17:35
• @Saad To your second comment: Yes, the piston pushes back according to Newton's 3rd law. But if the force needed to compress the gas is way larger, then the push from the piston doesn't do much. It will have some small effect and give a small pressure against the gas while it expands, but this is very small if the piston is not too heavy. Apr 24, 2017 at 17:37
• @Saad The second part of the question is better for a new question. Apr 24, 2017 at 17:38

The gas is providing pressure (and thus force). On a microscopic level you have lots of molecules moving in random directions. If there was no piston the gas would expand. Without a wall to supply a force the particles keep on moving. When there is a piston the particles bounce off the wall. Newton's third law applies for every little particle that bounces. The net effect of all these particles bouncing is a pressure. Gasses and liquids apply equal pressure in all directions so the gas is applying pressure to the wall as wel as it self.

The equation $$\Delta W=P\Delta V$$ with $$\Delta W$$ the work, $$P$$ the pressure and $$\Delta V$$ the change in volume holds for all types of materials as long as the change is not too abrupt. Otherwise the pressure can't be defined consistently. This equation is equivalent to $$\Delta W=F\Delta x$$ but applies to gasses/fluids. You can see this because pressure is force per area: $$\Delta W=\frac F A\cdot\Delta x\cdot A=P\Delta V$$

Let's consider one molecule that hits the piston. The kinetic energy of that molecule is supplied by breaking a chemical bond during combustion. Just before the molecule hits the piston the molecule's momentum is p = mv, and momentum of piston is p = 0. After the collision the law of conservation of momentum says that sum of momentum must be the same. Hence, as molecule bounce back with new momentum, the piston moves forward with some new momentum GREATER than zero. Hence the piston movement. With many of molecules hitting the piston it can move and produce useful work. For piston as a system the force comes from the molecule as external force and in accordance to Newton's Second Law piston will accelerate.