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# How to Calculate the force given potential energy

Any conservative force acting on an object within a system equals the negative derivative of the potential energy of the system with respect to x.

For one dimensional motion, the force can be found from Potential energy using following formula

$F_{x}=-\frac{\partial U}{\partial x}$

The generalized equation in three dimension is

$F_{x}=-\frac{\partial U}{\partial x}$
$F_{y}=-\frac{\partial U}{\partial y}$
$F_{z}=-\frac{\partial U}{\partial z}$

In vector form

$\boldsymbol{\mathbf{F}}=F_{x}\mathbf{i}+F_{y}\mathbf{j}+F_{z}\mathbf{k}$

Few examples to check on these
(1) Spring :
In the case of the deformed spring
$U=\frac{1}{2}Kx^{2}$
Now
$F_{x}=-\frac{\partial U}{\partial x}$
or
$F_{x}=-kx$

Which we already know is the restoring force in Spring mass system

(2)  Gravity

$U=mgH$
Now
$F_{x}=-\frac{\partial U}{\partial x}$
or
$F_{x}=-mg$
Which we already know is the gravitational force in gravity

(3) Potential Energy of a certain object is given by
$U= 10x^2 + 25z^3$
Now
$F_{x}=-\frac{\partial U}{\partial x}$
or
$F_{x}=-20x$
Also
$F_{y}=-\frac{\partial U}{\partial y} =0$
Also
$F_{z}=-\frac{\partial U}{\partial z} =-75z^2$
Hence the Force will be given
$\boldsymbol{F}=-20x \mathbf{i} -75z^2 \mathbf{k}$

(4) Potential Energy of a certain object is given by
$U= \frac {2yz}{x}$
Now
$F_{x}=-\frac{\partial U}{\partial x}$
or
$F_{x}= \frac {2yz}{x^2}$
Also
$F_{y}=-\frac{\partial U}{\partial y} =-\frac {2z}{x}$
Also
$F_{z}=-\frac{\partial U}{\partial z} = -\frac {2y}{x}$
Hence the Force will be given
$\boldsymbol{F}= \frac {2yz}{x^2} \mathbf{i} – \frac {2z}{x} \mathbf{j} – \frac {2y}{x} \mathbf{k}$

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