Here is the sketched situation: We have a bloc of mass M, and a spring attached to one end of that bloc. The bloc is moving and moves into a wall. The spring is compressed up until a certain maximum point. There is no friction between the bloc and the floor.

I have to justify why mechanical energy is conserved if we consider the system bloc&spring together. Usually, mechanical energy is conserved if there are only conservative forces applied to a system. But the wall's reaction force isn't a conservative force -- yet mechanical energy is clearly conserved, with the kinetic energy of the bloc transferred into potential energy for the spring.

What is the right justification for this?

  • $\begingroup$ The force is conservative. $\endgroup$ Nov 22 '16 at 23:23
  • $\begingroup$ The wall's reaction force is conservative? Ah, that would explain a lot. But then again, how do we justify that this force is conservative? It definitely works because it's parallel, not perpendicular, to the movement, right? $\endgroup$
    – el-flor
    Nov 22 '16 at 23:28
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    $\begingroup$ How much work is done by the wall? $W=F \times d$ $\endgroup$
    – BowlOfRed
    Nov 22 '16 at 23:30
  • $\begingroup$ The wall reaction force can be both conservative in its origin, if the wall is elactic and very hard, or not, e.g., static friction force, if the wall is actually some large stone lying on the ground. In any case, the work done by its force is negligible when the wall does not move, thus it does not affect energy conservation. $\endgroup$ Nov 22 '16 at 23:31
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    $\begingroup$ You got it. So a real wall could have some give, and d would not be zero, and the force would not be conservative. So you are asking the right question-- the answer is simply that we choose to idealize the situation in this way. We have to use a very rigid wall, or we are wrong. In the real world, you often have to look for the reasons that your idealizations didn't work! $\endgroup$
    – Ken G
    Nov 23 '16 at 1:40

This situation is no different than you assuming that an elastic ball hits a wall and rebounds elastically. In this case you spring is the spring like property of the bonds between the atoms of the ball and the wall.

So your system is the wall(with the Earth attached to it?) and the ball.
When you apply conservation of momentum (no external forces) and conservation of kinetic energy (elastic collision) you will find that the speed of the ball after rebound will be slightly less than the speed before rebound.
This is because the wall has got momentum and hence kinetic energy.

However when you compare the relative magnitudes of the masses of the wall and the ball can easily show that the rebound speed of the ball is equal to the incident speed of the ball to a very good approximation.
So the mechanical energy of the ball (spring and ball) system is conserved.


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