If there is container with a moveable piston. If heat is supplied to it the gas in container expands right raising the volume. The gas expands raises the piston against gravity (the gas does positive work on the weight). according to Newton's third law the gas exerts force PA on the piston but the piston also exerts and equal and opposite force. My question is how will piston even move if they exerts equal and opposite forces on one another.
The second question is that in above case if we calculate work done on the gas it comes out to be $W=\int( -PA)\;dx$. How? In my physics book they are saying that minus sign is due to the fact that force exerted by the piston on the gas is in opposite direction to displacement. But in many other places i have seen that work done formula is given by
$W = \int(PA)\;dx$. How?
Best Answer
I will answer the second question first. Work is a dot product. $W=\vec{F}\cdot \vec{x}$. This is works out nicely if $\vec{x}$ is fixed and $\vec{F}$ is not dependent on $x$. In your problem, $\vec{x}$ is indeed fixed but technically $\vec{F}$ is not. For the solution we must now do an integral $\int_{x_o}^{x_f} \vec{F}(x)\cdot d\vec{x}$. Let's look at your problem outside with physically what is happening. The work here is done by the gas. As the gas expands, a force is applied to the movable piston. The piston is moved in the direction of the force. So the dot product is positive and there is work done by the gas. The negative sign in the math comes in because the asked question is the work done on the gas.
Think of work as that thing which transforms energy from one form to another. In this example you increase the temperature of the gas. This gives the gas more energy. This increased energy is shown in the system as in increase of pressure of the gas. This is a force applied to the piston which has the force of gravity also applied. These two forces are for the moment not balanced. So work is done by the gas on the piston so that the forces to cancel out. And therefore some energy will move from the gas(in the form of pressure) into the gravitational potential energy of the piston.
Now as for Newton's third law. It is not actually correct. There are many examples where the third law does not hold. Where there is acceleration occurring, be weary of trusting the third law. Don't get me wrong, in static situations, the third law is a great tool and should be used. But in dynamics is can usually be thrown away.