Let's consider a block on a frictionless table. The block is connected to a fixed support on the table via a massless spring. Suppose the block is pulled aside by a distance x and then released. The potential energy of the "block+spring" system at the instant of release is taken to be 1/2 kx2 (provided potential energy is zero at x=0).
I am aware of the following.
If external forces do no work on the system and internal forces are conservative, the mechanical energy of the system remains constant.
The difference in potential energy corresponding to a conservative force is defined as negative of the work done by the force.
Now, the forces internal to the "spring+block" system are:
- Force by the spring on the block
- Force by the block on the spring
Why isn't the potential energy corresponding to the force 2 is not taken into account while computing the potential energy of the entire "system"?
Is is that force 2 is nonconservative?
My question may seem wierd. But when the gravitational potential energy of a body+earth system is defined, both works (by the gravitational force of the body and by the gravitational force of the earth) are taken into account. Work due to gravitational force of body is dismissed only because of its small value.
Best Answer
Here is a force diagram of your horizontal spring-mass system and its surroundings.
The two sets of forces $F_{\rm s,b}$ and $F_{\rm b,s}$, and $F_{\rm s,te}$ and $F_{\rm te,s}$ are Newton third law pairs, ie equal in magnitude and opposite in direction.
Making the assumption that the mass of the table and Earth is much greater that the mass of the block we can then assume that the right hand end of the spring does not move.
In your system only the block can have kinetic energy because the spring has no mass, and only the spring can have potential energy as the block is assumed rigid and there are no other forces as a result of an interaction between the block and the spring.
Suppose the block and the left hand end of the spring move together by a small distance $\Delta x$ to the right.
The work done on the block by the spring is $F_{\rm b,s}\,\Delta x$ (which increases the blocks kinetic energy) and the work done on the spring by the block is $-F_{\rm s,b}\,\Delta x$ (which decreases the potential energy stored in the spring).
The net amount of work done by those two internal forces is zero but it has resulted in an increase in the kinetic energy of the block and a corresponding decrease in the potential energy of the spring with no change in the total mechanical energy of the spring-mass system.