Class 11PhysicsWork, Energy, Power and CollisionConservative and Non-Conservative forces and Potential Energy
DifficultProve $F = - \frac{dV}{dx}$ for a conservative force.
(N/A) Suppose that a body undergoes a small displacement $dx$ under the action of a conservative force $F$.
The work done $dW$ by this conservative force is given by $dW = F dx$.
According to the work-energy theorem,the work done by all forces is equal to the change in kinetic energy,$dW = dK$.
From the law of conservation of mechanical energy,the total mechanical energy $E = K + V$ remains constant for a conservative force,meaning $dE = 0$.
Therefore,$dK + dV = 0$,which implies $dK = -dV$.
Substituting $dW = dK$ and $dW = F dx$,we get $F dx = -dV$.
Thus,$F = -\frac{dV}{dx}$.
Hence,for a conservative force,the force is equal to the negative gradient of the potential energy with respect to displacement.
The work done $dW$ by this conservative force is given by $dW = F dx$.
According to the work-energy theorem,the work done by all forces is equal to the change in kinetic energy,$dW = dK$.
From the law of conservation of mechanical energy,the total mechanical energy $E = K + V$ remains constant for a conservative force,meaning $dE = 0$.
Therefore,$dK + dV = 0$,which implies $dK = -dV$.
Substituting $dW = dK$ and $dW = F dx$,we get $F dx = -dV$.
Thus,$F = -\frac{dV}{dx}$.
Hence,for a conservative force,the force is equal to the negative gradient of the potential energy with respect to displacement.
Explore More
Similar Questions
$A$ body falls towards the Earth in air. Will its total mechanical energy be conserved during the fall? Justify.
Medium
View SolutionThe points of maximum and minimum attraction in the curve between potential energy $(U)$ and distance $(r)$ of a diatomic molecule are respectively:
Medium
View Solution$A$ particle moves in a potential region given by the formula $U = K(x^2 + y^2 + z^2)$. Find the force acting on the particle.
Easy
View SolutionThe potential energy of an object is $U(x) = (5x^2 - 4x^3) \ J$, where $x$ is the position in meters. The position at which the force becomes zero is
Given below are two statements:
Statement $I$: An object moves from position $r_{1}$ to position $r_{2}$ under a conservative force field $\vec{F}$. The work done by the force is $W = -\int_{r_{1}}^{r_{2}} \vec{F} \cdot d\vec{r}$.
Statement $II$: Any object moving from one location to another location can follow an infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below:
Statement $I$: An object moves from position $r_{1}$ to position $r_{2}$ under a conservative force field $\vec{F}$. The work done by the force is $W = -\int_{r_{1}}^{r_{2}} \vec{F} \cdot d\vec{r}$.
Statement $II$: Any object moving from one location to another location can follow an infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below:
Vedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo