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Why is it ($-U$) and not ($+U$)? And how do we know it's going to help us if we take it to be negative beforehand?

 Image from the Feynman lectures

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    $\begingroup$ It is in general better to type out the question rather than just posting a screenshot. Regardless, some resources on the sign convention for potential energy are here physics.stackexchange.com/questions/tagged/… $\endgroup$
    – G.Lang
    Commented Dec 28, 2020 at 17:00

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The negative is just a convention that essentially makes the total mechanical energy $E=K+U$ rather than $E=K-U$.

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  • $\begingroup$ Thanks for the explanation $\endgroup$ Commented Dec 28, 2020 at 17:07
  • $\begingroup$ But what problems would we face if it were K - U ? Just curious! $\endgroup$ Commented Dec 28, 2020 at 17:17
  • $\begingroup$ Probably knowledge of definite integral and boundary conditions will help in understanding the fact of Negative signs. $\endgroup$ Commented Dec 28, 2020 at 17:51
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Potential energy function sign depends on exact potential form definition. Usual convention is to use such sign of potential energy, so that field accomplished work by moving a particle from higher potential energy area to lower potential energy area, should be positive. So for example gravitational potential energy near Earth surface is defined as : $$ E = \pmb+mgh $$,

So that work done by gravity field is $$ W = E_2 - E_1 = mg\,(h_2 - h_1) \gt 0 $$

Now if two masses are separated apart by huge distances, then gravitational potential takes universal form of :

$$ U = \pmb- \frac {\alpha}{r} $$

So, for example Earth done work by attracting moon closer from distance $r_2$ to $r_1$ is :

$$ W = U_2 - U_1 = \alpha \left(\frac 1r_1 - \frac 1r_2\right) \gt 0$$, again positive work done by gravity.

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