Question

Is potential energy is equal to $ - \dfrac{{dU}}{{dx}}$ ?

Answer 1

Hint: First understand what potential energy is then try to write the equation of potential energy Compare it with the equation given in the question. Find out the derivative of potential energy .Try to equate both equations

Complete answer:
Potential energy is considered as the function of position. Potential energy is the work done in moving an object from one position to another. When an object is forced to move from a position ${x_1}$ to ${x_2}$ , the work done in moving that object will be equal but opposite to the force acting on that object.
Work Done $W = - \int\limits_{{x_1}}^{{x_2}} {F.dx} $ where F is the force and $dx$ is the change in position.
So from the equation it is clear that potential energy U is equivalent to work done
$\Rightarrow U = - \int\limits_{{x_1}}^{{x_2}} {F.dx} $
While taking derivatives on both sides of the above equation, we get
$\Rightarrow \dfrac{{dU}}{{dx}} = - F$
While comparing this with our question it is contradictory. Force equals the derivative of potential energy.
Potential energy is not equal to $ - \dfrac{{dU}}{{dx}}$ .
Negative sign in the equation of force indicates that work is done against the force.
For example Potential energy of a spring is given by
$\Rightarrow U = \dfrac{1}{2}K{x^2}$ , where \[K\] denotes the spring constant and x denotes the distance.
$\Rightarrow F = - \dfrac{{dU}}{{dx}} = - Kx$(\[F\] is the restoring force)
Now from the example it is clear that potential energy is not equal to $ - \dfrac{{dU}}{{dx}}$
$\Rightarrow F = - \dfrac{{dU}}{{dx}}$

Note: Try to write proper equations for potential energy and work done. From the equations try to derive the equation given in the question. Analyze whether the given equation matches with the obtained result. If it is contradictory, conclude by saying both the terms are not equal.