Question

The relation between the conservative force and the potential energy U is given by
A) $\overrightarrow{F}={\displaystyle \frac{dU}{dx}}$
B) $\overrightarrow{F}=\int Udx$
C) $\overrightarrow{F}=-{\displaystyle \frac{dU}{dx}}$
D) $F={\displaystyle \frac{dU}{dx}}$

Answer 1

Hint
To find the relation between conservative force and the potential energy use the formula by assuming the object at two different points-
$W_{ab}=U_a-U_b$
Or simply it can be also said as “Work done is equal to the negative change in the potential energy of the particle”.

Complete step-by-step answer:
Considering an object whose mass is $m$ and placed at point ‘$a$’ at height ‘$h_1$’ and the work done required to move the point from another point ‘B’ at height ‘$h_2$’. As we know that potential at point ‘a’ is given by $mg(h_1)$ or “ $U_a$”, and potential energy at point ‘B’ is given by $mg(h_2)$ or “$U_b$”.
And hence the work done in moving the particle from point ‘a’ to ‘b’ is given by
$W_{ab}=U_a-U_b$,
Or it can be also written as-
$W_{ab}=-(U_b–U_a)$ ………………(1)
Or it can be also define as-
The force due to the conservative field $mg$ is downward and in the opposite direction to the direction in which the mass is move upward a distance $h$ so the work done by the force due to conservative field is ‘$-mgh$’ and that by the definition is minus the change in the potential energy.
 $-(U_b–U_a)$ can also be written in the form $-{\displaystyle \frac{dU}{dx}}$
Hence equation (1) can also be written as-
$\overrightarrow{F}=-{\displaystyle \frac{dU}{dx}}$
Potential energy of the particle is equal to the negative of the work done by the conservative force.

Note
A conservative force is a force with the property that the total work done in moving a particle between the two points is independent of the path taken. Gravitational force is an example of the conservative force.