a. If the force on a particle at some point in space is zero, must its potential energy also be zero at that point? Explain.

b. If the potential energy of a particle at some point in space is zero, must the force on it also be zero at that point? Explain.

The following formula describes the relationship between force and potential energy at every point in space:

$F=-\frac{dU}{dr}$

Here $F$is the force at any point, $\frac{dU}{dr}$potential energy rate of change with regard to position$\mathit{r}$.

No, a zero force at any place indicates that the rate of change of potential energy energy with respect to position $\left(\frac{dU}{dr}\right)$is zero at that point, but the potential energy at that point cannot be determined. It is impossible to determine the amount of potential energy without first establishing a reference position. As a result, a zero force at any position does not imply that its potential energy is zero.

No, a zero potential energy at any point in space does not imply that the force is zero; the potential energy zero point is chosen at random. If the rate of change of the potential energy with respect to position $\left(\frac{dU}{dr}\right)$ is nonzero, there will be a force.