# Work and energy: conceptual doubt

Suppose I am sliding a block very slowly on a rough surface. If the block has travelled $$d$$ distance then work done by me is $$W_1=\mu mg d$$ and that by friction is $$W_2=-\mu mg d$$.

Now the energy transferred from me to block is $$\mu mgd$$ and that taken by friction from block is $$\mu mgd$$, The net energy of block remains same but the energy taken by friction evolves as heat and that is equal to my chemical energy consumed, so total energy of $$block + me$$ system remains constant.

Now If I pull a block of mass $$m$$ slowly towards up to a height $$h$$, then work done by me is $$W_1=mgh$$ (assuming $$h$$ is much less than radius of earth) and that by gravity is $$W_2=-mgh$$. Therefore $$mgh$$ goes from me to block and $$mgh$$ from block to earth, So here also energy of block doesn't change, then why do we say that potential energy of block increases.

I know I am lacking something here, as the total energy of the system would not be conserved if the block's energy doesn't change and my energy decreases.

So here also energy of block doesn't change, then why we say that potential energy of block increases.

This is a common confusion and is due to poor communication, not a failure in your understanding which appears to be correct.

Potential energy is energy that is available by virtue of the configuration of a system. In this case the potential energy is $$mgh$$. Now, consider what system this potential energy describes. We have $$m$$, which is a property of the block, but $$g$$ is a property of the earth, and $$h$$ is a relationship between the block and the earth. So the system that has potential energy $$mgh$$ is the system of both the block and the earth.

So, when we say "the potential energy of the block increases", we are actually making a mistake. We should say "the potential energy of the block+earth system increases". Thus, when you say:

Therefore $$mgh$$ goes from me to block and $$mgh$$ from block to earth

If we taught potential energy correctly from the beginning you would have said "Therefore $$mgh$$ goes from me to block and the block+earth system's potential energy is increased by $$mgh$$"

• thankyou for answering , but I got maximum part of your answer but still I have one confusion left . The earth does $-mgh$ work on block also and therefore takes $mgh$ energy from it so it's energy increases which is not true I think Commented Jul 29, 2021 at 13:18
• @lalittolani Remember, the earth and the block are part of the same system. So the earth does not do -mgh work on the block. The earth+block system do -mgh work on you. If you really want to consider the earth and the block as separate things then you actually have to consider the earth, the block, and the gravitational field. The earth doesn't move or change so it does no work. The gravitational field does -mgh work on the block, so the potential energy of the gravitational field has increased by mgh. The gravitational potential energy (mgh) belongs to the gravitational field in this view
– Dale
Commented Jul 29, 2021 at 14:44
• I cannot understand how earth does not do -mgh work on block. The earth+block system do work on you Commented Jul 29, 2021 at 14:57
• @lalittolani it seems like you need a discussion rather than just a straight Q&A, but this forum actively suppresses discussions. I encourage you to go to physicsforums.com where I am also active and where discussions are encouraged.
– Dale
Commented Jul 29, 2021 at 15:16
• Sorry , by mistake I landed at some other site :-\ (physicshelpforum.com), You may now check me on physicsforum.com Commented Jul 30, 2021 at 15:07

I beleive one aspect of your understanding that is incorrect is that of potential energy.

So here also energy of block doesn't change , then why we say that potential energy of block increases.

The term "potential energy of the block" is actually incorrect. Potential energy is a property of a system of two or more particles, in this case being that of the Earth and the block. It follows that the correct way of expressing this change in gravitational potential energy is that the gravitational potential energy of the Earth-block system increases.

Now for the reason of it's increase: You have exerted a force and the displacement of the point of application of force is not perpendicular to the force (or zero), which means that you have done work on the system comprising the Earth and the block. You have correctly written it as $$W_{\text ext} = mgh$$ However, if you include yourself as a part of the block + Earth system, the total energy is conserved. In such a case, the work that you would do would no longer be external to the system. All of the increase in potential energy of the block-Earth pair would be accompanied by a decrease in chemical potential energy in your body. This was also the source of the $$W_{ext}$$ had you not been a part of the system:$$\Delta U_{\text chem} + \Delta U_g = 0$$ And this shows that the total energy of the block+you+Earth system is infact always conserved.

Hope this helps.

• Thankyou for replying, I got what you want to say but please see my comment on @Dale 's answer Commented Jul 29, 2021 at 13:19

For both the sliding and lifting examples the net work done on the block is zero. All that means is that the change in the macroscopic kinetic energy (change in kinetic energy of the motion of the block as a whole) of the block is zero, per the work energy theorem. For the sliding example the velocity is constant so the change in kinetic energy is zero. For the lifting example presumably the block starts at rest (on the ground) and ends at rest at $$h$$, for a change in KE of zero.

But the work energy theorem does not preclude a change in potential energy because, as noted in other answers, potential energy is not a property of the object (the block) but a system property. That's because potential energy is the energy of the position of an object relative to something else. In this case, gravitational potential energy is a property of the Earth-block system due to the position of the block relative to the Earth.

Hope this helps.

• Yes your answer helps me but the work -mgh is done by earth so it takes mgh amount of energy from block, does this increases the earth's energy Commented Jul 29, 2021 at 21:47
• @lalittolani Yes. But the increase in gravitational potential energy is that of the Earth-block system, not the Earth alone nor the block alone. In the absence of either GPE would not exist. Commented Jul 29, 2021 at 22:06
• I got confused , you say that earth's energy increase but @Dale 's comment says something a little different Commented Jul 29, 2021 at 22:09
• I said the potential energy belongs to the combination of the block and earth not the earth alone or the block alone. It is a system property. I don't think that conflicts with what Dale says, though we may say it a little differently. Commented Jul 29, 2021 at 22:21
• Yes I understand it now as property of a combined system , not of a single object of the system but I only not understand that we do mgh work on block and increase the potential energy of earth+block but I cannot account explanation for -mgh work done by earth Commented Jul 30, 2021 at 5:20

See, the concept of potential energy there actually implies the work done by the gravitational forces. We introduce the potential energy for conservative forces specially for not calculating the work of these forces again every time. So in the second case you just choose whether you consider the work of the gravitational force or the potential energy, not both.