Medium
Obtain the relation between potential energy and force for a conservative force.
(N/A) For a conservative force $F$,the work done $W$ by the force during a small displacement $\Delta x$ is given by $W = F \Delta x$.
According to the work-energy theorem,the work done by all forces equals the change in kinetic energy,$\Delta K = W$.
Thus,$\Delta K = F \Delta x$.
From the law of conservation of mechanical energy,the sum of changes in kinetic energy and potential energy is zero: $\Delta K + \Delta V = 0$.
Substituting $\Delta K = F \Delta x$ into the equation,we get $F \Delta x + \Delta V = 0$.
Rearranging the terms,we have $F \Delta x = -\Delta V$.
Therefore,$F = -\frac{\Delta V}{\Delta x}$.
In the limit $\Delta x \to 0$,this becomes $F = -\frac{dV}{dx}$.
Thus,for a conservative force,the force is the negative gradient of the potential energy with respect to displacement.
According to the work-energy theorem,the work done by all forces equals the change in kinetic energy,$\Delta K = W$.
Thus,$\Delta K = F \Delta x$.
From the law of conservation of mechanical energy,the sum of changes in kinetic energy and potential energy is zero: $\Delta K + \Delta V = 0$.
Substituting $\Delta K = F \Delta x$ into the equation,we get $F \Delta x + \Delta V = 0$.
Rearranging the terms,we have $F \Delta x = -\Delta V$.
Therefore,$F = -\frac{\Delta V}{\Delta x}$.
In the limit $\Delta x \to 0$,this becomes $F = -\frac{dV}{dx}$.
Thus,for a conservative force,the force is the negative gradient of the potential energy with respect to displacement.
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