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I have a lot of doubts regarding the potential energy definitions

First of all,I would try to express my Understandings(they might be wrong)regarding the issue

I was told that if Work done on a body by a Force F is positive then The source Applying The force Looses some of its energy and that energy is transferred to the Body In the form Of kinetic energy

This means that Change in body's kinetic energy should be equal to work done(as stated By Work Energy Theoram)

But then A new definition was taught to me for example

Consider the system of A body 'P' and the earth,the Body 'P' of mass 'M' was kept on the ground surface

My understanding:: If I apply a Force F=mg on the body and lift it up A height H(velocity at the end is also 0) and Then I did A work On the body Which equals F•H,meaning I transferred energy of amount FH But at the same time Gravitational force was acting on the particle,hence Earth's Force Mg did a negative work on the body MgH,this means the earth(who was applying force)took out the energy out of the Body P that equals MgH(same amount of the energy that that my hands gave the body P

So net energy gained by the body=0

So I can't understand from where does the potential energy come from?why it is defined as work done by conservative force

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  • $\begingroup$ @ Dheeraj energy is more like an abstract mathematical currency. Sometimes just used as mathematical way to constrain the object, ie. saying it cannot create energy using the definition we have used for what is energy. $\endgroup$
    – James
    Oct 20 '21 at 13:56
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So I can't understand from where does the potential energy come from?

The potential energy comes from the negative work done by gravity. The work done by gravity is negative because the direction of the force is opposite to the displacement. When a force does negative work on an object it takes energy away from that object. In this case, gravity removes energy you gave the object due to your positive work, and stores it as gravitational potential energy of the earth-object system.

The work you do on the object is positive because your force is in the same direction as the displacement. If the negative work done by gravity exactly equals the positive work you did, the net work is zero. This would occur if the object began at rest on the ground and you brought it to rest at the height $h$. Since the object begins and ends at rest, the change in its kinetic energy ($\Delta KE$) is zero and from the work energy principle that means the net work done is zero.

Now, since gravity took the energy $mgh$ away from the object, what did it do with it? It stored it as a change in gravitational potential energy ($\Delta U$) of the earth-object system. Note that potential energy $U$ is a system property. The object alone does not posses it and the earth alone does not posses it. It only exists when both the object and the earth are present. Thus it "belongs" to both.

Suppose that instead of you brining it to rest at $h$, you kept lifting it so that it had a vertical velocity $v$ and kinetic energy of $\frac{1}{2}mv^2$ at $h$ . In that case at $h$ there would be a change in potential energy ($\Delta U$) and a change in kinetic energy ($\Delta KE$). The $\Delta KE$ then represents the net work done on the object per the work energy principle, or

$$W_{net}=W_{you}+W_{gravity}=\Delta KE$$

why it is defined as work done by conservative force

Gravity is a conservative force because it is a force whose work is determined only by the final displacement of the object acted upon. In this case, the vertical displacement of the object. Note that it didn't matter whether the object had a velocity at $h$. The work done by gravity is still $-mgh$. The work that you did, on the other hand, was different if you brought the object to rest at $h$ than if you gave it a velocity at $h$. It didn't only depend on the vertical displacement. Therefore the force you applied is non-conservative. We can now rewrite the above equation as

$$W_{net}=W_{non-conservative}+W_{conservative}=\Delta KE$$

Since the negative work done by gravity only equals the change in potential energy

$$W_{conservative}=-\Delta U$$

Hope this helps.

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  • $\begingroup$ I got my concept clear but a last doubt,Why does the Energy belongs to earth+object system,as earth has only done the work ,it should be possesed by Earth only? $\endgroup$ Oct 20 '21 at 16:56
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    $\begingroup$ @DheerajGujrathi If the object were not there, what potential energy would the earth own? Nothing. If the object were there but the earth was not, what potential energy would the object own? Nothing. The only thing the object owns is its KE, if any, and that would be do to you, an agent external to the object-earth system $\endgroup$
    – Bob D
    Oct 20 '21 at 17:11
  • $\begingroup$ where is that energy stored inside the system,in in the earth,or in the block? $\endgroup$ Oct 20 '21 at 17:22
  • $\begingroup$ Once again, it is stored in the system. All forms of potential energy are system properties. Please read this: britannica.com/science/potential-energy $\endgroup$
    – Bob D
    Oct 20 '21 at 17:36
  • $\begingroup$ Ohh,thanks for the link,So,finally,does it mean that Earth Did some negative work On the Body to Store the energy in the system,as that the system itself can use that energy(potential) to convert it to kinetic energy? $\endgroup$ Oct 20 '21 at 17:43
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The thing is that the work done by gravity is the potential energy.

Potential is just a name given to work done by conservative forces since their work isn't wasted but becomes "stored" (there is a potential for releasing it again).

You can thus call them work, $W_\text{conservative}$, or you can call them potential energies, $U$, depending on what you want. They are two sides of the same coin: $$W_\text{conservative}=-\Delta U.$$

So the work-energy theorem can be written: $$W=W_\text{non-conservative}+W_\text{conservative}=\Delta K$$ which shows that the work done by conservative forces must be equal to all other work done if there is no change in the speed, in the kinetic energy. Lift a box from the floor, for instance, and gravity (a conservative force) does the same negative work as you do positive work on the box.

If we in the work-energy theorem choose to use $U$ instead of $W_\text{conservative}$, then we have:

$$W=\Delta U+\Delta K.$$

With no change in kinetic energy, we see that the work done when lifting a box is converted straight into potential energy. This corresponds with the general energy conservation law (in this case for mechanical forces).

So, just keep in mind that work by conservative forces and potential energy are two ways to express the same thing.

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  • $\begingroup$ After The discussion,I got A lot of information,as well as different Modes of understandings,thanks $\endgroup$ Oct 20 '21 at 18:06

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