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Mechanics
Conservative Forces in Mechanics
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 of the particle.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of conservative and non-conservative forces, path independence of work done, relation between conservative force and potential energy, conservation of mechanical energy for point masses.
PGT extension
Mathematical condition for conservative forces using vector differential operators (curl of force field is zero), scalar potential function derivation, central forces as conservative systems, Lagrangian and Hamiltonian mechanics foundations.
Prerequisites
Line integrals in vector calculus
Work-energy theorem
Newton's laws of motion
Definition and Path Independence
A force F is conservative if the line integral of F along any path between two points A and B is independent of the path.
Mathematically expressed as the work done along a closed loop is identically zero: oint F • dr = 0.
Path independence implies that no mechanical energy is dissipated as heat or sound during the transit.
Curl-free Condition (Differential Criterion)
A necessary and sufficient condition for a force field to be conservative in a simply connected domain is that its curl vanishes identically.
Expressed mathematically as del × F = 0.
This implies that partial derivatives of force components satisfy exactness conditions, such as (∂F_x)/(∂y) = (∂F_y)/(∂x).
Relation with Potential Energy
A conservative force can be expressed as the negative gradient of a scalar potential energy function U(r).
Mathematically given by $F = -del U$, linking spatial rate of change of potential energy to force.
The change in potential energy ΔU between two points is defined as the negative of the work done by the conservative force: Δ$U = -W_{AB}$.
Conservation of Mechanical Energy
In a system where only conservative forces do work, the total mechanical energy $E = K + U remains constant$.
Any increase in kinetic energy K is accompanied by an equivalent decrease in potential energy U.
Non-conservative forces like friction violate this conservation by converting mechanical energy into thermal energy.
Derivations
Path Independence to Zero Closed-Loop Work
Conditions: Force field is conservative and continuous
Consider two paths Path 1 and Path 2 connecting points A and B in a force field.
By definition of conservative force, work done along Path 1 (W_1) equals work done along Path 2 (W_2): int_{A}^{B} (Path 1) F • d$r = \int_{A}^{B} (Path 2) F • dr$.
Reverse Path 2 to go from B back to A, such that work done along the reverse path is -W_2.
Combine the forward journey along Path 1 and the return journey along reversed Path 2 to form a closed loop: oint F • d$r = \int_{A}^{B} (Path 1) F • dr + \int_{B}^{A} (Path 2) F • dr = W_1 - W_2 = 0$.
Result: oint F • dr = 0
Force as Negative Gradient of Potential Energy
Conditions: 1D conservative system
Define work done by a conservative force moving a particle from x to x + dx as dW = F(x) dx.
By definition, work done equals the negative change in potential energy: d$W = -dU$.
Equate the two expressions: F(x) d$x = -dU$.
Solve for force by differentiating potential energy with respect to position: F(x) = -dU/dx.
Result: $F = -\frac{dU}{dx}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Work Done by Conservative Force
$$W_{AB} = \int_{A}^{B} F \cdot dr = -\Delta U = U_A - U_B$$
$W_AB = Work done$, U_A, $U_B = Potential energies at points A$ and B Joule (J)
Only conservative forces acting
Differential Conservative Criterion (Curl)
$$\nabla \times F = 0$$
F = Force vector field, \nabla = Del operator N/m^2
Differentiable vector field in simply connected region
Draw: A curve of Potential Energy U versus position x, showing minima, maxima, and point of inflection.
Interpret: At minima (stable equilibrium), dU/dx = 0 and d^2U/dx^2 > 0, so force F = 0 points back to equilibrium. At maxima (unstable equilibrium), d^2U/dx^2 < 0.
Exam use: Direct questions on finding equilibrium positions and stability criteria.
Work Done Along Different Paths
Draw: Two points A and B on a Cartesian plane connected by multiple curved and straight paths (Path 1, Path 2, Path 3) under a conservative force.
Interpret: Work done integrated along Path 1 equals Path 2 and Path 3, demonstrating path independence.
Exam use: Conceptual verification of conservative property in theoretical problems.
Conservative vs Non-Conservative Closed Loop Work
Draw: A circular closed path on an F-r plane where a conservative force traces a loop, and a friction force trace on the same loop.
Interpret: For conservative force, area enclosed (total work in loop) is zero. For friction, total work in loop is negative and non-zero.
Exam use: Distinguishing conservative forces from dissipative forces graphically.
Solved numericals
TGT core
A force field is given by $F = (3x^2 + 6y)\hat{i} + 6x\hat{j}$. Show that this force is conservative and find the corresponding potential energy function U(x, y), assuming U(0,0) = 0.
Check the exactness condition: $F_x = 3x^2 + 6y$ and $F_y = 6x$.
Calculate partial derivatives: (∂F_x)/(∂y) = 6 and (∂F_y)/(∂x) = 6.
Since (∂F_x)/(∂y) = (∂F_y)/(∂x), the force is conservative.
Use $F_x = -\partial U/\partial x \Rightarrow \partial U/\partial x = -(3x^2 + 6y)$.
Integrate with respect to x: U(x,y) = -x^3 - 6xy + g(y).
Differentiate U with respect to y and equate to -F_y: -∂U/∂$y = 6x - g'(y) = 6x \Rightarrow g'(y) = 0 \Rightarrow g(y) = C$.
Using U(0,0) = 0, C = 0. Therefore, U(x,y) = -x^3 - 6xy.
Answer: U(x, y) = -x^3 - 6xy
PGT extension
A particle of mass 2 kg moves in a conservative force field described by potential energy U(x) = x^3 - 3x Joules. Find the positions of equilibrium and determine their stability.
Find the force or condition for equilibrium: $F = -dU/dx = 0$.
Differentiate U(x): dU/d$x = 3x^2 - 3$.
Set dU/d$x = 0: 3x^2 - 3 = 0 \Rightarrow x^2 = 1 \Rightarrow x = 1 m$ and $x = -1 m$.
Test stability using the second derivative: $d^2U/dx^2 = 6x$.
Answer: x = 1 m (Stable equilibrium), $x = -1 m (Unstable equilibrium)$
PGT extension
Verify if the force vector $F = yz\hat{i} + xz\hat{j} + xy\hat{k}$ is conservative by evaluating its curl.
Write the curl expression: \nabla \times F = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ yz & xz & xy \end{vmatrix}.
Calculate the i-component: \frac{\partial}{\partial y}(xy) - \frac{\partial}{\partial z}(xz) = x - x = 0.
Calculate the j-component: \frac{\partial}{\partial z}(yz) - \frac{\partial}{\partial x}(xy) = y - y = 0.
Calculate the k-component: \frac{\partial}{\partial x}(xz) - \frac{\partial}{\partial y}(yz) = z - z = 0.
Since \nabla \times $F = 0\hat{i} + 0\hat{j} + 0\hat{k}$, the force field is conservative.
Answer: Force is conservative since \nabla \times F = 0
Assumptions and limitations
The domain of the force field must be simply connected for the vanishing curl condition to guarantee path independence.
Common misconceptions
Common error: All central forces are necessarily conservative.
Correct: Central forces are conservative only if their magnitude depends solely on the distance r from the center, i.e., $F = f(r)r_hat$.
Velocity-dependent central forces, such as magnetic Lorentz forces or drag forces directed towards a center, are non-conservative.
Common error: Friction is a conservative force because work done in a round trip is zero.
Correct: Friction is non-conservative; work done by friction along a closed loop is always negative and non-zero.
Friction always opposes motion, continuously dissipating mechanical energy into thermal energy.
Quick revision
A force is conservative if and only if oint F • d$r = 0 or \nabla \times F = 0$.
Work Done by Conservative Force: $W_{AB} = \int_{A}^{B} F \cdot dr = -\Delta U = U_A - U_B$
Differential Conservative Criterion (Curl): \nabla \times F = 0
Must remember
The line integral of a conservative force around any closed loop is identically zero: $\oint F \cdot dr = 0$.
A force field is conservative if and only if its curl vanishes identically in a simply connected domain: $\nabla \times F = 0$.
A conservative force is related to potential energy via the negative gradient operator: $F = -\nabla U$.
Total mechanical energy $E = K + U$ remains conserved if only conservative forces do work.
Friction and viscous drag are classic examples of non-conservative forces that dissipate mechanical energy into heat.
Glossary
Conservative ForceA force for which work done in moving an object between two points is independent of the path taken.
CurlA vector operator that describes the infinitesimal rotation of a vector field, mathematically $\nabla \times F$.
Potential EnergyStored energy of position relative to a zero configuration, related to force by $F = -\nabla U$.
Exactness ConditionThe mathematical requirement $\frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}$ for a 2D force field to be conservative.
Conservative vs Non-Conservative Forces
Parameter
Conservative Force
Non-Conservative Force
Path Independence
Independent of path
Depends on path
Work in Closed Loop
Zero ($\oint F \cdot dr = 0$)
Non-zero (dissipative)
Mechanical Energy
Conserved
Converted to heat/sound
Mathematical Test
$\nabla \times F = 0$
$\nabla \times F \neq 0$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following conditions is necessary and sufficient for a force field $F$ to be conservative in a simply connected domain?
Explanation: A force field is conservative if and only if its curl vanishes identically, i.e., $\nabla \times F = 0$.
2. The work done by a conservative force along a closed loop is:
Explanation: By definition, the line integral of a conservative force over any closed path is zero: $\oint F \cdot dr = 0$.
3. How is a conservative force $F$ related to its potential energy function $U$?
Explanation: A conservative force is expressed as the negative gradient of the potential energy function: $F = -\nabla U$.
4. If the components of a 2D force are $F_x$ and $F_y$, the exactness condition for it to be conservative is:
Explanation: The exactness condition derived from $\nabla \times F = 0$ in two dimensions is $\frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}$.
5. Which of the following is a non-conservative force?
Explanation: Friction is dissipative and non-conservative as it converts mechanical energy into thermal energy and depends on the path.
6. The total mechanical energy $E = K + U$ of a system is conserved if:
Explanation: Conservation of mechanical energy holds strictly when only conservative forces perform work within the system.
7. What is the change in potential energy $\Delta U$ defined as in terms of work done $W_{AB}$ by a conservative force?
Explanation: The change in potential energy between two points is defined as the negative of the work done by the conservative force: $\Delta U = -W_{AB}$.
8. The elastic potential energy stored in an ideal spring of stiffness $k$ stretched by $x$ is given by:
Explanation: Integrating Hooke's law force $F = -kx$ yields the elastic potential energy $U = \frac{1}{2}kx^2$.
9. Central forces are conservative if their magnitude depends solely on:
Explanation: A central force is conservative if it is a function of distance alone, i.e., $F = f(r)\hat{r}$.
10. If a force field is given by $F = y \hat{i} + x \hat{j}$, is it conservative?
Explanation: Here $F_x = y$ and $F_y = x$. Since $\frac{\partial F_x}{\partial y} = 1$ and $\frac{\partial F_y}{\partial x} = 1$, they are equal, confirming it is conservative.
11. Gravitational potential energy near the Earth's surface for a mass $m$ at height $h$ is:
Explanation: Near the Earth's surface where $g$ is constant, gravitational potential energy is given by $U = mgh$.
12. Which statement best describes path independence?
Explanation: Path independence means the line integral depends solely on the endpoints A and B, not on the trajectory taken.
13. What is the unit of the curl of a 3D force field in SI units?
Explanation: Since curl involves spatial derivatives ($\nabla \times$), its units are force per unit area, i.e., N/m$^2$ (or N/m for 2D depending on formulation).
14. In a conservative force field, if kinetic energy increases by $15\text{ J}$, what happens to the potential energy?
Explanation: Total mechanical energy is conserved ($K + U = \text{constant}$), so an increase in $K$ requires an equal decrease in $U$.
15. Which of the following central forces is NOT conservative?
Explanation: The magnetic Lorentz force depends on particle velocity ($F = q(v \times B)$), making it velocity-dependent and non-conservative.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A force field is given by $F = (2xy + z^3)\hat{i} + (x^2) \hat{j} + (3xz^2) \hat{k}$. Is this force conservative?
4. A body moves from point A to point B under the action of a conservative force. If the work done by the force is $40\text{ J}$, the change in potential energy $\Delta U = U_B - U_A$ is:
Explanation: Work done $W_{AB} = -\Delta U$. Since $W_{AB} = 40\text{ J}$, $\Delta U = -40\text{ J}$.
5. Which of the following vector fields represents a possible conservative force field?
Explanation: For $F = x\hat{i} - y\hat{j}$, $\frac{\partial F_x}{\partial y} = 0$ and $\frac{\partial F_y}{\partial x} = 0$, so exactness holds. For others, partial derivatives do not match.
6. A particle of mass $m$ is dropped from a height $h$. Assuming mechanical energy conservation, its speed when it hits the ground is:
Explanation: Initial energy $mgh = \frac{1}{2}mv^2$, which gives $v = \sqrt{2gh}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The line integral $\oint F \cdot dr = 0$ for a closed path is a:
Explanation: By Stokes' theorem and vector calculus, zero work around every closed loop is both necessary and sufficient for a force to be conservative.
2. If $U = x^2 + y^2 + z^2$, the magnitude of the conservative force at point $(1, 2, 2)$ is:
3. Which of the following statements about non-conservative forces is correct?
Explanation: Non-conservative forces (like friction) are path-dependent, unlike conservative forces.
4. A particle moves under a conservative force $F$. If its potential energy decreases by $20\text{ J}$, its kinetic energy must:
Explanation: By conservation of mechanical energy ($K + U = \text{constant}$), a decrease in potential energy implies an equal increase in kinetic energy.
5. The potential energy of a conservative system is given by $U(x) = \frac{a}{x^2} - \frac{b}{x}$. For stable equilibrium, the value of $x$ is:
Explanation: For equilibrium, $\frac{dU}{dx} = -\frac{2a}{x^3} + \frac{b}{x^2} = 0 \implies \frac{2a}{x^3} = \frac{b}{x^2} \implies x = \frac{2a}{b}$. Second derivative test confirms stability.
6. What does the divergence of a conservative force field ($\nabla \cdot F$) typically represent?
Explanation: While curl vanishes for conservative forces ($\nabla \times F = 0$), divergence depends on the field sources (e.g., Gauss's law for gravity or electrostatics).
7. If a force field is conservative, the mixed partial derivatives of its potential function satisfy:
Explanation: Clairaut's theorem ensures that for smooth potential functions, mixed partial derivatives are equal: $\frac{\partial^2 U}{\partial x \partial y} = \frac{\partial^2 U}{\partial y \partial x}$.
8. Which of the following is true regarding central conservative forces?
Explanation: Since central forces act along the radius vector, torque $\tau = r \times F = 0$, meaning angular momentum is conserved.
9. A force $F = yz \hat{i} + xz \hat{j} + xy \hat{k}$ is:
Explanation: Check curl: $\nabla \times F$: $\partial_y(xy) - \partial_z(xz) = x - x = 0$, and similarly for other components. Thus, $\nabla \times F = 0$, making it conservative with potential $U = -xyz$.
10. If non-conservative forces (like air resistance) do negative work on a moving body, its total mechanical energy:
Explanation: Work-energy theorem for non-conservative forces states $W_{nc} = \Delta E_{mech}$. Negative work by non-conservative forces causes total mechanical energy to decrease.