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Mechanics
Non-Conservative Forces
Non-conservative forces are interactions where the work done in moving a particle between two points depends on the specific path taken. In the presence of such forces, mechanical energy is not conserved and typically dissipates into thermal or internal energy.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Identification of conservative vs non-conservative forces, basic work-energy theorem application with friction, and simple numericals on work done by sliding friction.
PGT extension
Path dependence of line integrals, mathematical conditions for non-conservativity (curl F not equal to zero), velocity-dependent forces like viscous drag, and Lagrangian formulations with Rayleigh dissipation functions.
Prerequisites
Work, Energy and Power
Conservative Forces and Potential Energy
Line Integrals in Vector Calculus
Definition and Path Dependence
A force F is non-conservative if the line integral of the force around any closed path is non-zero: ∮ F • dr ≠ 0.
The work done by a non-conservative force depends on the trajectory or path connecting the initial and final positions.
Examples include kinetic friction, viscous drag, and air resistance.
Violation of Mechanical Energy Conservation
Total mechanical energy $E = K + U$ is not conserved when non-conservative forces act on a system.
The work-energy theorem states that the total work done by all forces equals the change in kinetic energy: $W_net = \DeltaK$.
Writing $W_net = W_c + W_nc$, where W_c is conservative and W_nc is non-conservative work, leads to $W_nc = \DeltaK + \DeltaU = \DeltaE$.
Velocity Dependence and Dissipation
Many non-conservative forces, such as fluid drag ($F = -bv or F = -cv^{2})$, depend explicitly on the velocity of the particle.
These forces invariably oppose motion, continuously removing mechanical energy from the system and converting it into heat or sound.
Because they depend on velocity, they cannot be derived from a scalar potential energy function U(r).
Mathematical Test for Conservativity
For a force field to be conservative, its curl must vanish identically throughout the region: ∇ × F = 0.
If ∇ × F ≠ 0, the force field possesses circulation and is strictly non-conservative.
Forces like the magnetic Lorentz force or frictional forces fail the curl test or are velocity-dependent in ways that preclude a scalar potential.
Derivations
Work Done by Sliding Friction along Different Paths
Conditions: A block of mass m is moved on a horizontal rough surface with coefficient of kinetic friction μ_k.
Consider moving a block from point A(0,0) to B(L,0) along a straight path directly along the x-axis. The normal force is N = mg, and the frictional force is $f_k = \mu_k mg pointing opposite$ to displacement. The work done is $W_1 = \int (-\mu_k mg) dx from x = 0$ to L, yielding $W_1 = -\mu_k mg L$.
Now consider moving the same block along a semicircular path of radius R from A(0,0) to B(πR,0) in the xy-plane, where πR = L. At every infinitesimal displacement ds, the frictional force opposes the tangential motion: f_k • ds = -f_k ds.
Integrating the friction force along the arc length yields $W_2 = - \int \mu_k mg ds = -\mu_k mg \int ds = -\mu_k mg (\piR)$. Since πR > L, |W_2| > |W_1|, demonstrating path dependence.
Result: $W_friction = -\mu_k mg S$, where S is the total actual path length traversed.
Work-Energy Theorem with Non-Conservative Forces
Conditions: A particle moves under the combined action of conservative internal forces and external non-conservative forces.
Start with Newton's second law: F_ne$t = F_c + F_nc = m (dv/dt)$, where F_c is conservative and F_nc is non-conservative.
Take the dot product of the equation with displacement differential dr = v dt and integrate from initial position 1 to final position 2: ∫ (F_c • dr) + ∫ (F_nc • dr) = ∫ m (dv/dt) • v dt.
Recognize that the work done by conservative forces is related to potential energy change: ∫ F_c • dr = -Delta U, and the right-hand side integrates to kinetic energy change ΔK.
Rearrange the terms to express the non-conservative work: $W_nc = \DeltaK - (-\delta U) = \DeltaK + \DeltaU = \DeltaE$.
\Delta $E_{cycle} = mechanical energy lost in one complete closed cycle$ Joule (J)
Cyclic motion in a dissipative medium
Graphs and experiments
Work vs Path Length for Friction
Draw: A linear graph plotting Work Magnitude (|W|) on the y-axis against Path Length (s) on the x-axis, showing a straight line passing through the origin with slope μ_k mg.
Interpret: The work magnitude increases linearly with the distance traversed, highlighting that longer paths suffer greater energy dissipation.
Exam use: Tested in conceptual MCQs asking why friction cannot have a potential energy function.
Mechanical Energy Decay with Time
Draw: An exponential decay curve plotting Total Mechanical Energy E(t) on the y-axis against time t on the x-axis for an oscillator under damping.
Interpret: Energy is monotonically lost due to non-conservative damping forces until the system reaches rest.
Exam use: Used to interpret damped harmonic oscillation questions in PGT mechanics.
Joule's Paddle Wheel Experiment
Draw: Schematic setup showing falling weights connected via pulleys to a paddle wheel immersed in an insulated container of water, with a thermometer.
Interpret: Demonstrates the mechanical equivalent of heat by showing that work done by non-conservative gravitational/tension forces converted via friction/viscosity raises water temperature.
Exam use: Theoretical linkage between mechanical work done by non-conservative forces and internal thermal energy generation.
Solved numericals
TGT core
A block of mass 2 kg is pulled across a rough horizontal floor by a horizontal force over a distance of 5 m. The coefficient of kinetic friction between the block and the floor is 0.3. Find the work done by the frictional force. (Take $g = 10 m/s^2)$
Identify the given values: mass m = 2 kg, distance s = 5 m, coefficient of kinetic friction \m$u_k = 0.3$, $g = 10 m/s^2$.
Calculate the normal force N = mg = 2 \times 10 = 20 N.
Calculate the magnitude of kinetic friction $f_k = \mu_k N = 0.3 \times 20 = 6 N$.
Calculate the work done by friction: $W_f = -f_k s = -6 \times 5 = -30 J$.
Answer: -30 J
PGT extension
A particle of mass 1 kg moves in a horizontal xy-plane under a non-conservative force given by F = (-y i + x j) N. Calculate the work done by this force along a quarter circle of radius R = 2 m from (2, 0) to (0, 2).
Express the position vector and its differential along the circular path: x = R cos(\theta), y = R sin(\theta), where R = 2.
The displacement differential is dr = dx i + dy j. In polar coordinates or parametric form: d$x = -R \sin(\theta) d\theta$ and dy = R cos(\theta) d\theta.
Substitute into the force vector: F = (-R sin(\theta) i + R cos(\theta) j).
Compute the dot product F • dr = (-R sin(\theta))(-R sin(\theta) d\theta) + (R cos(\theta))(R cos(\theta) d\theta) = R^2 (sin^2(\theta) + cos^2(\theta)) d$\theta = R^2 d\theta$.
Integrate from \theta = 0 to $\theta = \pi/2: W = \int_0^{\pi/2} R^2 d\theta = R^2 [\theta]_0^{\pi/2} = (2^2)(\pi/2) = 4(\pi/2) = 2\pi J$.
Answer: 2\pi J (approx 6.28 J)
PGT extension
A small body of mass 0.5 kg falls from rest through a viscous medium that exerts a drag force F_$d = -bv$, where $b = 2 Ns/m$. Find its velocity as a function of time t. (Take $g = 9.8 m/s^2)$
Set up Newton's second law along the vertical downward direction (y-axis): mg - b$v = m (dv/dt)$.
Rearrange into a standard linear first-order differential equation: dv/dt + (b/m)v = g.
Integrate both sides using the initial condition v(0) = 0: v(t) = (9.8 / 4) (1 - e^{-4t}) = 2.45 (1 - e^{-4t}) m/s.
Answer: v(t) = 2.45(1 - e^{-4t}) m/s
Assumptions and limitations
Macroscopic friction is assumed to obey Amontons-Coulomb laws where the kinetic friction coefficient is independent of sliding speed and apparent contact area, which breaks down at extremely high velocities or nanocontacts.
Viscous drag formulations assume specific flow regimes (such as linear Stokes drag or quadratic Newton drag) and fail when turbulence or compressibility effects become prominent.
Common misconceptions
Common error: Friction always does negative work, so total energy of the universe decreases.
Correct: Friction does negative mechanical work, converting mechanical energy into thermal energy; total energy remains conserved.
Energy is conserved overall; non-conservative forces only transfer mechanical energy into internal thermal energy modes.
Common error: If the work done by a force between two points is zero, the force must be conservative.
Correct: Zero work along one specific path does not guarantee zero work along all paths or zero work around a closed loop.
Conservativity requires zero work for *every* arbitrary closed path, not merely a specific round trip.
Common error: Non-conservative forces cannot be expressed as analytical functions of position or velocity.
Correct: Forces like F = -bv or position-dependent non-conservative circulating forces are fully expressed via analytical vector functions.
An analytical formula can define a non-conservative vector field perfectly; the distinguishing feature is the non-zero curl or path dependence.
Quick revision
Non-conservative forces exhibit path-dependent work and non-zero closed-loop line integrals: ∮ F • dr ≠ 0.
The presence of non-conservative forces implies that mechanical energy changes by exactly the work they perform: $W_nc = \DeltaE$.
Non-Conservative Work Line Integral: $W_{nc} = \oint \mathbf{F}_{nc} \cdot d\mathbf{r} \neq 0$
Must remember
The line integral of a non-conservative force around any closed path is non-zero: $\oint \mathbf{F}_{nc} \cdot d\mathbf{r} \neq 0$.
Total mechanical energy is not conserved in the presence of non-conservative forces; instead, $W_{nc} = \Delta E = \Delta K + \Delta U$.
Velocity-dependent forces like viscous drag ($F = -bv$) and kinetic friction are prime examples of non-conservative forces.
A force field is conservative if and only if its curl vanishes identically everywhere: $\nabla \times \mathbf{F} = 0$.
Non-conservative forces dissipate mechanical energy into non-mechanical forms such as internal thermal energy or sound.
Glossary
Non-Conservative ForceA force for which the work done depends on the path taken between two points and whose work around a closed loop is non-zero.
Dissipative ForceA type of non-conservative force (like friction or drag) that continuously removes mechanical energy from a system and converts it into heat.
Curl of a Vector FieldA mathematical operator ($\nabla \times$) measuring the rotation or circulation of a vector field; it must be zero for conservative force fields.
Stokes' TheoremA theorem relating the line integral of a vector field around a closed loop to the surface integral of its curl over any surface bounded by the loop.
Viscous DragA velocity-dependent resistive force experienced by objects moving through fluids, proportional to velocity at low Reynolds numbers.
Conservative vs Non-Conservative Forces
Feature
Conservative Forces
Non-Conservative Forces
Path Dependence
Independent of path (depends only on endpoints)
Depends explicitly on the trajectory or path
Closed Path Work
$\oint \mathbf{F} \cdot d\mathbf{r} = 0$
$\oint \mathbf{F} \cdot d\mathbf{r} \neq 0$
Potential Energy
Can be derived from a scalar potential $U(\mathbf{r})$
Cannot be derived from a scalar potential function
Mechanical Energy
Conserves total mechanical energy ($E = K + U = \text{constant}$)
Alters mechanical energy ($W_{nc} = \Delta E$)
Mathematical Test
$\nabla \times \mathbf{F} = 0$
$\nabla \times \mathbf{F} \neq 0$ or velocity-dependent
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following mathematical conditions must be satisfied for a force field $\mathbf{F}$ to be strictly conservative?
Explanation: A force field is conservative if and only if its curl vanishes identically everywhere throughout the region, i.e., $\nabla \times \mathbf{F} = 0$.
2. What is the work done by a non-conservative force around any closed path?
Explanation: By definition, a non-conservative force yields a non-zero line integral around a closed loop: $\oint \mathbf{F} \cdot d\mathbf{r} \neq 0$.
3. Which of the following is a classic example of a non-conservative force?
Explanation: Kinetic friction is dissipative and path-dependent, making it a non-conservative force. Gravity, electrostatics, and spring forces are conservative.
4. How does the total mechanical energy $E = K + U$ behave when non-conservative forces act on a system?
Explanation: The work-energy theorem with non-conservative forces states that $W_{nc} = \Delta K + \Delta U = \Delta E$, meaning mechanical energy is not conserved.
5. Why do velocity-dependent forces like fluid drag fail to possess a scalar potential energy function $U(\mathbf{r})$?
Explanation: Potential energy functions depend solely on spatial position coordinates $U(\mathbf{r})$. Forces explicitly dependent on velocity $\mathbf{v}$ cannot be expressed as a simple gradient of a position-only potential.
6. A block slides across a rough horizontal table over a distance $s$ under a constant kinetic friction force $f_k$. What is the work done by friction?
Explanation: Frictional force opposes displacement, so the work done is $W_f = -f_k s = -\mu_k mg s$, which is negative.
7. According to Stokes' theorem, the line integral of a force $\mathbf{F}$ around a closed contour is equal to:
Explanation: Stokes' theorem relates the circulation $\oint \mathbf{F} \cdot d\mathbf{r}$ directly to the surface integral of the curl of $\mathbf{F}$ over any bounded surface $S$.
8. What is the instantaneous power dissipated by a linear drag force $\mathbf{F}_d = -b\mathbf{v}$ acting on a particle moving with speed $v$?
Explanation: Power is given by $P = \mathbf{F}_d \cdot \mathbf{v} = (-b\mathbf{v}) \cdot \mathbf{v} = -bv^2$, representing energy loss per unit time.
9. If a force field is given by $\mathbf{F} = y\hat{i} + x\hat{j}$, is it conservative or non-conservative?
Explanation: Calculating the curl: $\nabla \times \mathbf{F} = \left(\frac{\partial}{\partial x}x - \frac{\partial}{\partial y}y\right)\hat{k} = 0$. Since the curl vanishes, the force field is conservative.
10. Which of the following statements regarding non-conservative forces is incorrect?
Explanation: Non-conservative forces (like friction) typically dissipate and decrease mechanical energy in an isolated system, rather than increasing it.
11. Stokes' Law gives the viscous drag force on a small sphere of radius $r$ moving at speed $v$ in a fluid of viscosity $\eta$ as:
Explanation: For low Reynolds number laminar flow, Stokes' Law states $F_d = -6\pi \eta r \mathbf{v}$.
12. What happens to the work done by a non-conservative force if the initial and final points of a trajectory are identical (closed loop)?
Explanation: Unlike conservative forces where closed-loop work is always zero, non-conservative forces yield a non-zero work value around any closed loop.
13. Which force among the following is velocity-dependent and non-conservative?
Explanation: The magnetic Lorentz force $\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$ depends on velocity and does no work, but is categorized among velocity-dependent forces that lack a standard scalar potential depending solely on position.
14. In the presence of both conservative and non-conservative forces, the total work $W_{net}$ done on a particle equals:
Explanation: The work-energy theorem states that net work done by *all* forces equals the change in kinetic energy: $W_{net} = \Delta K$.
15. Why is sliding friction considered path-dependent?
Explanation: The work done by friction depends directly on the total distance $s$ traversed along the path, making it path-dependent.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The work done by a non-conservative force around a closed path is:
Explanation: By definition, the line integral of a non-conservative force around any closed path is non-zero: $\oint \mathbf{F} \cdot d\mathbf{r} \neq 0$.
2. If a force field is conservative, which of the following vector differential relations holds true?
Explanation: A necessary and sufficient condition for a force field to be conservative is that its curl is identically zero everywhere: $\nabla \times \mathbf{F} = 0$.
3. A body of mass $m$ slides down a rough inclined plane of length $L$ and inclination $\theta$ with coefficient of kinetic friction $\mu_k$. The work done by friction is:
Explanation: The normal force on the incline is $N = mg \cos\theta$. The work done by friction over distance $L$ is $W_f = -\mu_k N L = -\mu_k mg L \cos\theta$.
4. Which of the following forces is non-conservative and velocity-dependent?
Explanation: Viscous drag force (e.g., $F = -bv$) depends explicitly on velocity and is a non-conservative dissipative force.
5. In a system where both conservative and non-conservative forces do work, the relationship between non-conservative work $W_{nc}$ and mechanical energy change $\Delta E$ is:
Explanation: Writing total work as $W_{net} = W_c + W_{nc} = \Delta K$, and noting $W_c = -\Delta U$, we get $W_{nc} = \Delta K + \Delta U = \Delta E$.
6. A force field is given by $\mathbf{F} = yz\hat{i} + xz\hat{j} + xy\hat{k}$. This force field is:
Explanation: Calculating the curl: $\nabla \times \mathbf{F} = (\partial_y(xy) - \partial_z(xz))\hat{i} + \dots = (x - x)\hat{i} + (y - y)\hat{j} + (z - z)\hat{k} = 0$. Since curl is zero, it is conservative.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the physical significance of a non-zero line integral $\oint \mathbf{F} \cdot d\mathbf{r} \neq 0$?
Explanation: A non-zero closed-loop line integral indicates that mechanical energy is continuously dissipated or added over a cycle by non-conservative forces.
2. A particle moves in a medium with linear viscous drag $\mathbf{F} = -b\mathbf{v}$. If its initial speed is $v_0$, what is the instantaneous power dissipation at $t = 0$?
Explanation: Power dissipated is $P = \mathbf{F} \cdot \mathbf{v} = (-b\mathbf{v}_0) \cdot \mathbf{v}_0 = -bv_0^2$.
3. Which of the following force fields is definitely non-conservative?
Explanation: For $\mathbf{F} = y\hat{i} - x\hat{j}$, $\nabla \times \mathbf{F} = (0 - 0)\hat{i} + (0 - 0)\hat{j} + (-1 - 1)\hat{k} = -2\hat{k} \neq 0$. Thus, it is non-conservative.
4. If a block of mass $2\text{ kg}$ is pulled over a rough surface with $\mu_k = 0.2$ for a distance of $5\text{ m}$, what is the work done by kinetic friction? ($g = 10\text{ m/s}^2$)
5. Why cannot a scalar potential energy function $U$ be defined for frictional forces?
Explanation: A potential energy function requires a unique value at every point in space. Path dependence means work varies with trajectory, precluding a single-valued potential function.
6. Stokes' Theorem relates which two mathematical constructs?