How can the energy of a system increase even if net work done on it is zero?

Question

Imagine two blocks of mass 1kg placed on a horizontal frictionless surface. Now, if I apply 10 N on the right block in the right direction, and on the left block in the left direction, both the blocks will accelerate and gain velocity. This will result in increased kinetic energy of the blocks, thus result in an increase in the total energy of the system. But again, considering these two blocks to constitute a system, the net force on the system is zero, and yet the energy of the system is increasing. How is this possible? Because as far as I know,

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W = F.s

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But since net force is zero, how is work being done on the system? (That is, how is its internal energy increasing?)

Answer 1

In $W=\vec F \cdot \vec s$ the $\vec F$ is a force acting on the system and $\vec s$ is the displacement of the material of the system where the force is acting.

So in your example both forces are acting in the direction of the displacement of the material of the system at the point where each force is acting. So the work from each force is positive and the total work is therefore positive.

There is a separate quantity which superficially looks similar, but is conceptually distinct. It is $W_{net}=\vec F_{net}\cdot \vec s_{CoM}$. This quantity is called the “net work” because it is related to the net force, but it is not the total work done by all forces acting on the system. The net work is only the portion of the total work that changes the KE of the center of mass.

So in your example the net work is zero. Therefore the KE of the CoM does not change. All of the total work goes to increasing the internal energy.

Answer 2

Either the pair of blocks are one thing, in which case the one thing's kinetic energy is the kinetic energy associated with the translation of the center of mass of the pair of blocks, which doesn't move even though the blocks are moving... or the pair of blocks are two separate things, in which case there are two centers of mass with two displacements which need to be tracked separately.

If the two blocks are two things, you have done mechanical work on two things with two centers of mass, moving them along two paths, transferring energy from the chemical potential energy in your body to the translational kinetic energy of two blocks.

If the two blocks are one thing, you have increased that thing's thermal internal energy by thermodynamic work across the boundary, transferring energy from the chemical potential energy in your body to the thermal internal energy of the one thing. This is obviously a very odd way of looking at a system of two macroscopic particles being accelerated by two discrete external forces. But if we were talking about something like an explosion of fuel in a piston, it's a perfectly natural way to look at it: the chemical potential energy of the fuel-air mixture, which we exclude from our system, does thermal work by the application of countless electromagnetic force interactions on the particles which constitute the gas, increasing the thermal internal energy of the gas, but not accelerating its center of mass.

Answer 3

This will result in increased kinetic energy of the blocks, thus result in an increase in the total energy of the system.
is a correct statement but must include an extra word,
$\dots$ increase in the total internal energy of the system.

You should start by defining the system.
For simplicity assume that there is a constant force of attraction, magnitude $F$, between the two blocks which are of the same mass, initially separated and at rest.

System - block $1$
External force - force on block $1$ due to block $2$.
Block $1$ moves a distance $\Delta x$ and so the work done by external force on block $1$ is $F\,\Delta x$ and thus is equal to the increase in kinetic energy of block $1$.

System - block $2$
External force - force on block $2$ due to block $1$.
Block $2$ moves a distance $\Delta x$ and so the work done by external force on block $2$ is $F\,\Delta x$ and thus is equal to the increase in kinetic energy of block $2$.

System - block $1$ & block $2$
External force - none.
Work done on system by external force - zero, ie the centre of mass of the system does not move.

You can extend the analysis by changing the nature of the forces, masses of the block etc, but always you must differentiate between the internal and external forces.

Answer 4

The forces are equal and opposite, but the displacements are also equal and opposite. The displacement of each block is in the direction of the force on each, so the work done on each block is positive. The work done adds up not subtracts.