Question

prove that the gradient of potential energy is the measure of force​

Answer 1

NAZUAK❤

Because the gradient of any field (like the gravitational potential field) points in the direction of largest increase in that field. ... The force will point in the direction of the largest decrease in potential (energy) — in the negative of the gradient of the potential field.

Answer 2

Gradient of Potential Energy is FORCE !

• We can prove this with DIMENSIONAL ANALYSIS.

To prove :

$\boxed{ \bigg| \dfrac{dU}{dx} \bigg| = f}$

Proof:

LHS :

$\therefore \: \dfrac{dU}{dx}$

$= \: \dfrac{ \bigg \{U \bigg \}}{ \bigg \{x \bigg \}}$

$= \: \dfrac{ \bigg \{M{L}^{2} {T}^{ - 2} \bigg \}}{ \bigg \{L\bigg \}}$

$= \: \bigg \{M{L}^{1} {T}^{ - 2} \bigg \}$

RHS:

$\therefore \: \bigg \{f \bigg \}$

$= \: \bigg \{M{L}^{1} {T}^{ - 2} \bigg \}$

$= \: LHS$

So, LHS = RHS.

[Hence, proved]