Summary
This is Lecture 8 for the topic “Work, Energy and Power”. You can find all resources related with Work, Energy and Power by clicking on this link. Here we will cover the concept of Conservative and Non Conservative Forces
1. Introduction to Force Fields
In mechanics, when a particle moves through space, it may experience forces that depend on its position, velocity, or time. The nature of these forces dictates whether mechanical energy is conserved or dissipated. Forces are broadly categorized into two major classes: Conservative and Non-Conservative.
2. Conservative Forces
A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the taken path. It depends only on the initial and final positions.
Key Properties of Conservative Forces:
- Path Independence: The work done by a conservative force along path 1 from point $A$ to point $B$ is identical to the work done along path 2 between the same two points:
$$W_{AB} (\text{Path 1}) = W_{AB} (\text{Path 2})$$ - Work in a Closed Loop is Zero: If a particle starts at point $A$, moves along any arbitrary closed loop, and returns to $A$, the net work done by a conservative force is zero:
$$\oint \vec{F}_c \cdot d\vec{r} = 0$$ - Association with Potential Energy: A conservative force can always be expressed as the negative gradient of a scalar potential energy function $U$:
$$\vec{F} = -\nabla U = -\left( \frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k} \right)$$ - Conservation of Mechanical Energy: When only conservative forces do work, the total mechanical energy ($K + U$) remains constant.
The Mathematical Test for Conservative Forces (JEE Favorite):
For a two-dimensional force field $\vec{F} = F_x \hat{i} + F_y \hat{j}$, the force is conservative if and only if it satisfies the exactness condition (derived from curl $\nabla \times \vec{F} = 0$):
$$\frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}$$
Common Examples:
Gravitational force, ideal spring force ($F = -kx$),
electrostatic force ($F = \frac{kq_1q_2}{r^2}$).
3. Non-Conservative Forces
A non-conservative force is a force for which the work done depends explicitly on the path taken between two points.
Key Properties of Non-Conservative Forces:
- Path Dependence: If you take a longer, winding path against a non-conservative force (like friction), more energy is dissipated than if you take a straight path.
- Work in a Closed Loop is Non-Zero: Moving an object in a closed loop under a non-conservative force results in net non-zero work (usually negative, as energy is continuously drained):
$$\oint \vec{F}_{nc} \cdot d\vec{r} \neq 0$$ - Dissipative Nature: Non-conservative forces typically convert macroscopic mechanical energy into microscopic thermal energy, sound, or internal deformation energy. Mechanical energy is not conserved ($W_{nc} = \Delta E$).
- No Potential Energy Function: You cannot define a potential energy field $U$ for a non-conservative force.
Common Examples: Kinetic friction, static friction (when slipping occurs), air resistance, fluid viscosity, and tension from an active non-ideal string/motor.
4. Distinguishing Features Summary Table
| Feature | Conservative Forces | Non-Conservative Forces |
| Path Dependence | Independent of path (depends only on endpoints) | Dependent on the specific path taken |
| Closed Loop Work | $\oint \vec{F} \cdot d\vec{r} = 0$ | $\oint \vec{F} \cdot d\vec{r} \neq 0$ |
| Energy Conversion | Converts between Kinetic and Potential energy | Dissipates Mechanical energy into Thermal/Internal energy |
| Potential Energy | Can be defined ($U$ exists) | Cannot be defined |
| Mathematical Test | $\frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}$ (in 2D) | Fails the exactness derivative test |
| Examples | Gravity, Spring Force, Electrostatic Force | Friction, Air Drag, Viscous Force |
JEE-Mains/NEET Practice Questions
Question 1: Mathematical Test for Conservative Force
Which of the following force fields is conservative?
(a) $\vec{F} = (xy\hat{i} + x^2y\hat{j}) \text{ N}$
(b) $\vec{F} = (y^2\hat{i} + 2xy\hat{j}) \text{ N}$
(c) $\vec{F} = (x\hat{i} – y\hat{j}) \text{ N}$
(d) Both (b) and (c)
Solution: (d)
Test each option using the exactness condition $\frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}$:
- For (a): $F_x = xy \implies \frac{\partial F_x}{\partial y} = x$. $F_y = x^2y \implies \frac{\partial F_y}{\partial x} = 2xy$. Since $x \neq 2xy$, not conservative.
- For (b): $F_x = y^2 \implies \frac{\partial F_x}{\partial y} = 2y$. $F_y = 2xy \implies \frac{\partial F_y}{\partial x} = 2y$. Since $2y = 2y$, conservative.
- For (c): $F_x = x \implies \frac{\partial F_x}{\partial y} = 0$. $F_y = -y \implies \frac{\partial F_y}{\partial x} = 0$. Since $0 = 0$, conservative.
Therefore, both (b) and (c) are conservative force fields.
Question 2: Work in a Closed Loop
A block is slid along a rough horizontal floor in a complete circular path of radius $R$ and returned to its exact starting point under the action of kinetic friction. The total work done by kinetic friction is:
(a) Zero
(b) $-\mu mg (2\pi R)$
(c) $+\mu mg (2\pi R)$
(d) Dependent on the speed of the block
Solution: (b)
Kinetic friction is a non-conservative, dissipative force that always opposes motion. In a closed loop, friction never cancels out; it always performs negative work over every differential segment of the path ($d\vec{r}$).
$W = \oint (-f_k) ds = -f_k \times (\text{Circumference}) = -\mu mg (2\pi R)$.
Question 3: Conceptual Identification
Assertion (A): The work done by the gravitational force in moving a particle around any closed path is zero.
Reason (R): Gravitational force is a conservative force.
(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
(b) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(c) (A) is true, but (R) is false.
(d) (A) is false, but (R) is true.
Solution: (a)
By definition, one of the primary defining characteristics of a conservative force (like gravity) is that the net work done in any closed path is strictly zero. Thus, Reason (R) directly explains Assertion (A).
Question 4: Potential Energy Existence
A force is given by $\vec{F} = (3x^2 y^2 \hat{i} + 2x^3 y \hat{j}) \text{ N}$. Find the change in potential energy ($\Delta U$) of the system when a particle moves from the origin $(0,0)$ to the point $(1, 2)$.
Solution:
First, check if it’s conservative:
$F_x = 3x^2y^2 \implies \frac{\partial F_x}{\partial y} = 6x^2y$
$F_y = 2x^3y \implies \frac{\partial F_y}{\partial x} = 6x^2y$
Since partial derivatives match, the force is conservative, meaning potential energy exists.
$\Delta U = -W_c = – \int_{(0,0)}^{(1,2)} (F_x dx + F_y dy)$
Notice that $F_x dx + F_y dy = 3x^2y^2 dx + 2x^3y dy$ is the exact differential of $U(x,y) = x^3 y^2$.
Integrating from $(0,0)$ to $(1,2)$:
$U(1,2) = (1)^3 (2)^2 = 4 \text{ J}$.
$U(0,0) = (0)^3 (0)^2 = 0 \text{ J}$.
Change in potential energy $\Delta U = U_f – U_i = 4 – 0 = 4 \text{ J}$. (Thus, work done by the conservative force is $-4 \text{ J}$).