This may sound trivial but I wanted to make sure I did not misunderstand. I know that the conservation of energy is always true even in cases where energy appears to be lost (when in reality it is just converted into heat, sound etc.). However, my question is does the conservation of energy hold when there are unbalanced forces acting on an object? For example, does it make a difference in terms of the conservation of energy if I lift a book to a height of 1 m with acceleration (and therefore a net upwards force) or if I lift the book to 1 m at a constant velocity (no net force)? I would answer no, but the answer to the question says the conservation of energy is only true if all forces are balanced which I don't think is true. Could someone please let me know which is correct?
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$\begingroup$ Related: physics.stackexchange.com/q/19216/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Aug 5, 2020 at 23:27
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$\begingroup$ Forces are ALWAYS balanced per Newton's 3rd law. For accelerating objects, the work/kinetic-energy theorem is applicable. $\endgroup$– David WhiteCommented Aug 6, 2020 at 1:53
4 Answers
Energy is conserved in the whole universe *. Nevertheless, if you delimit a system and you focus only on that system, forces can vary the energy of the system. Total energy is conserved in the universe, but in your particular system migh not.
If you wide your system to include more objects, then you'll find energy conserved.
[*] Energy will be conserved as long as time is translationally symmetric (i.e. all instants are equivalent), according to Noether's theorems.
To be clear, if you "lift a book at constant velocity" you are still having to provide a force against gravity and so are still "doing work". Perhaps what you mean is an object moving at constant velocity in a vacuum. If no external forces act on this object its energy doesn't change.
If you're asking if the conservation of energy holds in the two following situations in a vacuum:
You apply a force $F$ to an initially stationary object and move it from $A$ to $B$.
You allow an object to travel at a constant velocity between $A$ and $B$ without applying any force to it.
then the answer is yes. Energy is conserved in both of these cases.
In the first case you (the person pushing the object) are applying a force to the object and giving up some of your energy which turns into kinetic energy for the object. The total energy you lose to the object equals the total kinetic energy gained by the object. So no net energy is being lost, only transferred.
In this situation energy is neither being lost nor gained by both you and the object.
When you lift a book you are expending energy, you can accelerate it all the time you are lifting it, or you can lift it steadily at a constant speed. In both cases you are increasing the book's gravitational potential energy, also some of your energy may go to air friction and sound waves. So none of the energy you spend is lost.
Yes, the conservation of energy holds when there are net forces too, and the difference between the two cases that you proposed is the behaviour of kinetic energy:
- if the net force is zero the speed is constant, as well as the kinetic energy; also consider that in reality -even when you try to lift a book at constant speed- the net force applied to the book (you + gravity) must still be non-zero when you start and you finish: it is initially positive (meaning that $F_{you}>F_g$) to get it moving from rest and negative to make it stop, in such a way that the total work stays equal to zero.
- in the accelerated case the final kinetic energy is greater than the initial kinetic energy, and their difference is exactly the "net" work of the forces, as described by the work-energy principle: $$ L = \Delta K $$
