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Mechanics
Conservation of Energy
The principle of conservation of energy states that the total mechanical energy of an isolated system remains constant if only conservative forces act upon it. It serves as a foundational conservation law derived from the time-translation invariance of physical systems.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Work-energy theorem, conservative and non-conservative forces, conservation of mechanical energy in one-dimensional motion, potential energy curves, and simple harmonic motion applications.
PGT extension
General conservative force fields defined by curl-free conditions (nabla × F = 0), potential energy in three dimensions, central force motion energy integrals, and work-energy relations in rotating or non-inertial frames involving fictitious forces.
Prerequisites
Newton's laws of motion and free-body diagrams
Line integrals of vector fields and scalar products
Concept of center of mass and external/internal forces
Conservative Forces and Potential Energy
A force F is conservative if the work done by it along a closed path is zero: oint F · dr = 0.
The work done by a conservative force is independent of the path taken and depends only on the initial and final positions.
Potential energy U(r) is defined exclusively for conservative forces such that delta $U = -W = -\int F \cdot dr$.
Mechanical Energy Conservation Law
Total mechanical energy E is the sum of kinetic energy (K) and potential energy (U): $E = K + U$.
When only conservative forces do work, delta $K = -\delta U$, which implies delta $E = \delta K + \delta U = 0$.
In the presence of non-conservative forces like friction (f_k), the change in mechanical energy equals the work done by non-conservative forces: delta $E = W_{nc} = -f_k s$.
Potential Energy Curves and Equilibrium
The force is related to the potential energy function by the negative gradient: F(x) = -dU/dx.
Stable equilibrium occurs at local minima of the potential energy curve where dU/dx = 0 and d²U/dx² > 0.
Unstable equilibrium occurs at local maxima where dU/dx = 0 and d²U/dx² < 0, while neutral equilibrium corresponds to a flat region.
Energy Conservation in Central Force Fields
Central forces (such as gravitational and electrostatic forces) are directed toward a fixed center and are inherently conservative.
For a particle in a central force field, angular momentum L is conserved alongside total energy E, reducing the effective 3D problem to 1D motion.
The effective potential energy includes the centrifugal barrier term: U_{eff}(r) = U(r) + L² / (2mu r²).
Derivations
Work-Energy Theorem to Conservation of Mechanical Energy
Conditions: Particle of mass m moving under the influence of a conservative force F(x)
Start with Newton's second law: $F = m(dv/dt) = m v (dv/dx)$.
Integrate the force from position x_i to x_f to find work done: $W = \int_{x_i}^{x_f} F(x) dx = \int_{x_i}^{x_f} m v (dv/dx) dx = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2 = K_f - K_i$.
By definition of potential energy for a conservative force, $W = U(x_i) - U(x_f) = - (U_f - U_i)$.
Equating the two expressions for work: K_f - $K_i = -(U_f - U_i)$, which rearranges to K_i + $U_i = K_f + U_f$.
Result: $E = K + U = \text{constant}$
Condition for a Force to be Conservative
Conditions: Vector force field F(x, y, z) acting in a simply connected region
Express the work done along an arbitrary path as $W = \int (F_x dx + F_y dy + F_z dz)$.
For work to be path-independent, the integrand must be an exact differential, implying F · d$r = -dU$.
Using Stokes' theorem on a closed loop, oint F · d$r = iint (\nabla \times F) \cdot n dA = 0$ for any surface bounded by the loop.
For this integral to vanish for all arbitrary surfaces, the integrand must vanish identically throughout the region.
Result: \nabla \times F = 0
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Work-Energy Theorem
$$W_{net} = \Delta K = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$$
$W_{net} = Net work done$, m = mass, $v_f = final velocity$, $v_i = initial velocity$ Joules (J)
Valid in both inertial and non-inertial frames if pseudo-forces are included
Relation Between Force and Potential Energy
$$F(x) = -\frac{dU}{dx}$$
F(x) = Conservative force component, U = Potential energy Newtons (N)
F = Force vector, \nabla = Gradient operator, U = Potential energy scalar field Newtons (N)
Conservative force field
Conservation of Mechanical Energy
$$K_1 + U_1 = K_2 + U_2$$
K = Kinetic energy, U = Potential energy at states 1 and 2 Joules (J)
Only conservative forces perform work
Gravitational Potential Energy (Near Earth)
$$U = mgh$$
m = mass, g = acceleration due to gravity, h = height above reference level Joules (J)
Uniform gravitational field near Earth's surface
Elastic Potential Energy of a Spring
$$U = \frac{1}{2}kx^2$$
k = spring constant, x = displacement from natural length Joules (J)
Ideal spring obeying Hooke's law
Effective Potential in Central Force Motion
$$U_{eff}(r) = U(r) + \frac{L^2}{2\mu r^2}$$
U(r) = Central potential, L = Angular momentum, \mu = Reduced mass, r = radial distance Joules (J)
Central force motion in polar coordinates
Total Mechanical Energy with Non-Conservative Forces
$$\Delta E = \Delta K + \Delta U = W_{nc}$$
$W_{nc} = Work done$ by non-conservative forces such as friction or drag Joules (J)
General mechanical systems with dissipation
Graphs and experiments
Potential Energy Curve U(x) vs Position x
Draw: A smooth curve with multiple local minima and maxima, showing a horizontal total energy line E intersecting the curve.
Interpret: Regions where E > U(x) represent allowed motion where kinetic energy is positive. Points where E = U(x) are turning points. Local minima indicate stable equilibrium; local maxima indicate unstable equilibrium.
Exam use: Questions asking to identify equilibrium types, turning points, and maximum velocity positions from graphical data.
Simple Pendulum Energy Conservation
Draw: A bar chart or stacked area plot showing kinetic energy and potential energy varying inversely with displacement, while their sum remains constant over one time period.
Interpret: At the extreme amplitudes, potential energy is at its maximum and kinetic energy is zero. At the mean position, kinetic energy is at its maximum and potential energy is at its reference minimum.
Exam use: Numerical and conceptual problems on finding string tension, velocity at the lowest point, and height of swing.
Mod Atwood Machine Energy Dissipation
Draw: A linear plot of total mechanical energy versus time for a system experiencing sliding friction, showing a negative slope proportional to the frictional dissipation rate.
Interpret: The decrease in mechanical energy per unit time directly equals the thermal energy generated by non-conservative friction forces.
Exam use: Distinguishing isolated conservative systems from dissipative systems in multi-body problems.
Solved numericals
TGT core
A particle of mass 2 kg moves in a conservative force field where the potential energy is given by U(x) = 3x^3 - 6x^2 Joules. Find the force acting on the particle at x = 2 m.
Recall the relation between force and potential energy in 1D: F(x) = -dU/dx.
Differentiate U(x) with respect to x: dU/d$x = d/dx (3x^3 - 6x^2) = 9x^2 - 12x$.
Substitute $x = 2 m into the derivative: dU/dx|_{x=2} = 9(2)^2 - 12(2) = 9(4) - 24 = 36 - 24 = 12 J/m$.
Calculate the force: F(2) = - (12) = -12 N.
Answer: -12 N
PGT extension
A 1 kg block slides down a frictionless track from a height of 5 m and then enters a rough horizontal surface with a coefficient of kinetic friction \m$u_k = 0.2$. How far will the block travel on the rough surface before coming to rest?
Apply conservation of mechanical energy for the descent to find velocity at the bottom: \frac{1}{2}m$v^2 = mgh \implies v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 5} = \sqrt{98} m/s$.
Use the work-energy theorem for the motion on the rough surface: \Delta $K = W_{nc}$.
The final kinetic energy is zero, so 0 - \frac{1}{2}m$v^2 = -f_k s$, where $f_k = \mu_k mg$.
Substitute the values: \frac{1}{2}m(2gh) = \mu_k mg s \implies mg$h = \mu_k mgs$.
Cancel mass m and solve for distance s: $s = \frac{h}{\mu_k} = \frac{5}{0.2} = 25 m$.
Answer: 25 m
PGT extension
A particle of mass m moves in a central potential U(r) = - \frac{k}{r}. If the particle moves in a circular orbit of radius r_0, find its total mechanical energy in terms of k and r_0.
The centripetal force is provided by the negative derivative of the potential: F(r) = -dU/d$r = -\frac{k}{r^2}$.
The potential energy at r_0 is U($r_0) = -\frac{k}{r_0}$.
Total energy $E = K + U = \frac{k}{2r_0} - \frac{k}{r_0} = -\frac{k}{2r_0}$.
Answer: -\frac{k}{2r_0}
Assumptions and limitations
Macroscopic energy conservation assumes macroscopic bodies where internal quantum states and rest-mass energy conversions remain constant.
Potential energy formulations require forces to be strictly position-dependent and curl-free (conservative); velocity-dependent forces like magnetic Lorentz forces or resistive drag cannot be fully represented by a scalar potential alone.
Common misconceptions
Common error: Friction always violates the conservation of energy law.
Correct: Friction converts mechanical energy into thermal energy, but total energy of the universe remains strictly conserved.
Mechanical energy is not conserved in dissipative systems, but the total energy including internal thermal energy is always conserved.
Common error: Potential energy belongs exclusively to a single object.
Correct: Potential energy is a property of the configuration of a system of interacting bodies.
Gravitational or electrostatic potential energy requires at least two interacting masses or charges.
Common error: Work done by a conservative force depends on the chosen path if the speed changes.
Correct: Work done by a conservative force depends solely on the initial and final positions, regardless of path or speed.
By definition, line integrals of conservative vector fields around any open path depend strictly on endpoints.
Quick revision
Mechanical energy $E = K + U$ is conserved if and only if all working forces are conservative.
Force is the negative spatial derivative of potential energy: $F = -dU/dx$.
Stable equilibrium exists at local minima of potential energy where d²U/dx² > 0.
Must remember
A force is conservative if the work done along a closed path is zero: $\oint F \cdot dr = 0$.
The relation between force and potential energy in 1D is given by $F(x) = -\frac{dU}{dx}$.
Stable equilibrium corresponds to local minima of the potential energy curve where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} > 0$.
In the presence of non-conservative forces like friction, the change in mechanical energy equals the work done by non-conservative forces: $\Delta E = W_{nc}$.
For a particle in a central force field, both total energy $E$ and angular momentum $L$ are conserved.
Glossary
Conservative ForceA force for which work done is independent of path and depends only on initial and final positions.
Potential EnergyEnergy associated with the configuration of a system of interacting bodies, defined as $\Delta U = -W$.
Effective PotentialThe combination of true potential and centrifugal potential in central force motion: $U_{eff}(r) = U(r) + \frac{L^2}{2\mu r^2}$.
Stable EquilibriumA state of minimum potential energy where a slight displacement results in a restoring force towards equilibrium.
Conservative vs Non-Conservative Forces
Feature
Conservative Force
Non-Conservative Force
Path Dependence
Independent of path
Dependent on path
Closed Path Work
Zero work done along any closed path
Non-zero work done along a closed path
Associated Potential Energy
Can be defined
Cannot be defined
Examples
Gravitational force, electrostatic force, spring force
Friction, air resistance, viscous drag
Practice Quiz
Choose an option to reveal the explanation.
1. What is the work done by a conservative force along a closed path?
Explanation: By definition, the work done by a conservative force along any closed path is zero: $\oint F \cdot dr = 0$.
2. Which of the following is a non-conservative force?
Explanation: Frictional force dissipates mechanical energy into thermal energy and depends on the path, making it non-conservative.
3. The relationship between a one-dimensional conservative force $F(x)$ and potential energy $U(x)$ is:
Explanation: A conservative force is related to potential energy as the negative gradient: $F(x) = -\frac{dU}{dx}$.
4. Stable equilibrium for a mechanical system occurs when:
Explanation: Stable equilibrium requires a local minimum of potential energy, which satisfies $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} > 0$.
5. What is the change in mechanical energy when only non-conservative forces like friction do work?
Explanation: The change in mechanical energy equals the work done by non-conservative forces: $\Delta E = W_{nc}$.
6. What is the expression for the effective potential energy in a central force field?
Explanation: The effective potential includes the centrifugal barrier: $U_{eff}(r) = U(r) + \frac{L^2}{2\mu r^2}$.
7. In a three-dimensional conservative force field, how is force expressed in terms of potential energy $U$?
Explanation: In 3D, a conservative force is the negative gradient of the scalar potential field: $F = -\nabla U$.
8. Unstable equilibrium is characterized by which of the following conditions?
Explanation: Unstable equilibrium corresponds to a local maximum of the potential energy curve, where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} < 0$.
9. Which physical quantity is conserved alongside total energy for a particle moving in a central force field?
Explanation: Central forces exert zero torque about the center of force, ensuring angular momentum $L$ is conserved.
10. The elastic potential energy stored in an ideal spring of spring constant $k$ stretched by $x$ is:
Explanation: Integrating Hooke's law force yields the elastic potential energy formula $U = \frac{1}{2}kx^2$.
11. What condition must be satisfied by a force field $F$ in three dimensions for it to be conservative (assuming simply connected domain)?
Explanation: A force field is conservative if and only if its curl is zero: $\nabla \times F = 0$.
12. What does neutral equilibrium correspond to on a potential energy curve?
Explanation: Neutral equilibrium occurs in a flat region where both first and second derivatives of $U$ vanish, meaning the particle remains in equilibrium wherever displaced.
13. The Work-Energy Theorem states that the net work done on an object equals:
Explanation: The work-energy theorem states $W_{net} = \Delta K = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$.
14. Gravitational potential energy near the Earth's surface for a mass $m$ at height $h$ is given by:
Explanation: In a uniform gravitational field near Earth's surface, gravitational potential energy is $U = mgh$.
15. Why can potential energy not be defined for a non-conservative force like kinetic friction?
Explanation: Potential energy requires work to depend only on endpoints. Since friction depends on the path, a unique potential energy function cannot be defined.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A particle moves in a potential field given by $U(x) = \frac{a}{x^2} - \frac{b}{x}$, where $a$ and $b$ are positive constants. The position of stable equilibrium is at:
Explanation: For equilibrium, $\frac{dU}{dx} = -\frac{2a}{x^3} + \frac{b}{x^2} = 0 \implies \frac{2a}{x^3} = \frac{b}{x^2} \implies x = \frac{2a}{b}$. Checking the second derivative $\frac{d^2U}{dx^2} = \frac{6a}{x^4} - \frac{2b}{x^3}$, substituting $x = \frac{2a}{b}$ yields a positive value, confirming stable equilibrium.
2. The potential energy of a particle of mass $m$ is given by $U(x) = \frac{1}{2}kx^2$ for $x < 0$ and $U(x) = 0$ for $x \ge 0$. If total mechanical energy is $E$, what is the maximum displacement to the left ($x < 0$)?
Explanation: At maximum displacement, kinetic energy is zero, so $E = U(x) = \frac{1}{2}kx^2$. Solving for $x$ when $x < 0$ gives $x = -\sqrt{\frac{2E}{k}}$.
3. A force $F = (2xy + z^3)\hat{i} + x^2\hat{j} + (3xz^2 - 2)\hat{k}$ acts on a particle. Is this force conservative?
Explanation: Calculating the curl of $F$: $\nabla \times F = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\hat{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\hat{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\hat{k} = (0 - 0)\hat{i} + (3z^2 - 3z^2)\hat{j} + (2x - 2x)\hat{k} = 0$. Since $\nabla \times F = 0$, the force is conservative.
4. A block of mass $m$ slides down a rough inclined plane of inclination $\theta$ and length $L$ with coefficient of kinetic friction $\mu_k$. The net change in mechanical energy of the block when it reaches the bottom is:
Explanation: The non-conservative force is friction $f_k = \mu_k N = \mu_k mg \cos\theta$. The work done by friction over distance $L$ is $W_{nc} = -f_k L = -\mu_k mg L \cos\theta$, which equals the change in mechanical energy $\Delta E$.
5. A particle of mass $m$ moves in a central potential $U(r) = \frac{1}{2}kr^2$. What is the angular frequency $\omega$ of circular orbits of radius $r_0$?
Explanation: The effective radial force for circular motion at equilibrium radius is given by $F_{eff} = -\frac{dU_{eff}}{dr} = 0$. Here $U(r) = \frac{1}{2}kr^2$, so the centripetal force condition gives $m \omega^2 r_0 = kr_0 \implies \omega = \sqrt{\frac{k}{m}}$.
6. If a conservative force is given by $F = -kx$, the potential energy function $U(x)$ taking $U(0) = 0$ is:
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the primary characteristic that distinguishes conservative forces from non-conservative forces?
Explanation: Conservative forces perform work that is independent of path, depending solely on initial and final positions.
2. A particle is placed in a potential energy well described by $U(x) = x^4 - 2x^2$. How many equilibrium positions does it have, and which are stable?
Explanation: $\frac{dU}{dx} = 4x^3 - 4x = 4x(x-1)(x+1) = 0$, giving equilibrium points at $x = 0, 1, -1$. The second derivative $\frac{d^2U}{dx^2} = 12x^2 - 4$. At $x = 0$, $\frac{d^2U}{dx^2} = -4$ (unstable). At $x = 1$ and $x = -1$, $\frac{d^2U}{dx^2} = 8 > 0$ (stable). Thus, there are 2 stable and 1 unstable equilibria.
3. Which of the following line integrals equals zero for any closed path if force $F$ is conservative?
Explanation: By definition, the work done around any closed path by a conservative force is zero: $\oint F \cdot dr = 0$.
4. In central force motion, the presence of the centrifugal term $\frac{L^2}{2\mu r^2}$ in the effective potential accounts for:
Explanation: The centrifugal barrier term $\frac{L^2}{2\mu r^2}$ incorporates the rotational kinetic energy due to angular momentum $L$, reducing the 3D problem to an effective 1D radial motion.
5. A body of mass $m$ is dropped from a height $h$ under gravity. What is its velocity just before hitting the ground, using energy conservation?
Explanation: By mechanical energy conservation, initial potential energy equals final kinetic energy: $mgh = \frac{1}{2}mv^2 \implies v = \sqrt{2gh}$.
6. If the potential energy of a system is given by $U = cxy$ where $c$ is a constant, the force component $F_x$ is:
Explanation: The force component $F_x = -\frac{\partial U}{\partial x} = -\frac{\partial}{\partial x}(cxy) = -cy$.
7. Which statement is correct regarding total energy conservation in the universe?
Explanation: While mechanical energy can dissipate into heat due to friction, the total energy of an isolated system (the universe) remains conserved according to the First Law of Thermodynamics.
8. What is the nature of equilibrium at a point where $\frac{dU}{dx} = 0$ and $\frac{d^2U}{dx^2} = 0$, but $\frac{d^3U}{dx^3} \neq 0$?
Explanation: When the first non-vanishing derivative occurs at an odd order (like third derivative), the point is a point of inflection and corresponds to unstable or neutral characteristics depending on context, specifically not a strict local extremum.
9. A spring with spring constant $k$ is stretched by distance $x_0$. The work done by an external agent in slowly stretching the spring is:
Explanation: The work done by an external agent is equal to the change in potential energy of the spring, which is $\frac{1}{2}kx_0^2$.
10. Under what condition is total mechanical energy conserved in a system?
Explanation: Total mechanical energy is conserved ($\Delta E = 0$) if and only if no non-conservative forces do work on the system.