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The potential energy can be mathematically expressed as:

$$U=-\int \vec F \cdot d\vec x$$

I do understand the negative sign is basically to adjust in the relation $$K_1+U_1=K_2+U_2$$

What is the physical significance of the negative sign.

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It means that traveling along a conservative force increases your kinetic energy at the expense of potential energy, and traveling against it decreases your kinetic energy and adding it to the potential energy.

The significance is in the negative work because a conservative force isn't associated with dissipative forces that waste energy.

If it does negative work and decreases your kinetic energy that energy is not wasted and is stored as potential energy

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  • $\begingroup$ The sign is just a convention. No dissipation is introduced by reversing the usual sign convention. $\endgroup$ Jun 5, 2021 at 10:01
  • $\begingroup$ @GiorgioP Doesn't it stem from our definition of potential energy as "Energy stored in a conservative force field"? Only by that we get dU=-Fdx afaik. $\endgroup$ Jun 5, 2021 at 10:36

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