How potential energy is equal to -dU/dx ?

1 Answer
MetaPhysik
Feb 2, 2018

They are not equal.

Explanation:

It is the force that is equal to #-(dU)/dx#

#F_x = - (dU)/dx#

where F_x is the force in x-diretion, and U is the potential energy. dU/dx is how potential energy in x-direction.

Example 1, for a spring system
#U = 1/2kx^2#

#rArr F_x = - (dU)/dx =- kx#

Obviously, #F_x# is the restoring force of the spring when it is compressed or stretched, it's direction of which is always opposite to the compression or extension.

Example 2, for gravity

#U= -G(m_1m_2)/r #

#rArr F_r = -(dU)/(dr) =G(m_1m_2)/r^2 #

Hence #F_r# is the restoring force tugging on a mass body, like an invisible spring.

A restoring force that are independent of pathways is also called a conservative force.

Finally, the U can come in many forms, many of them are energy stored in distorted shapes such a bow, a twisted or bent rod or from bodies that reside in force fields such a magnets (or magnetic dipoles) in magnetic fields, a charge in electric field etc etc.

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