Conservative Forces and Potential Energy: Complete Physics Guide with HRK Notes

House of Physics October 19, 2025
Conservative Forces and Potential Energy: Complete Physics Guide | B.Sc. Mechanics

Conservative Forces and Potential Energy: Complete Physics Guide

Mastering Conservative Forces, Potential Energy, Energy Conservation, and Equilibrium in Physical Systems
Conservative Forces Potential Energy Spring Force Gravitational Force Energy Conservation Reading Time: 25 min

📋 Table of Contents

📜 Historical Background

The concept of conservative forces and potential energy developed throughout the 17th-19th centuries:

  • Galileo Galilei (1638): Early work on energy conservation in falling bodies
  • Gottfried Wilhelm Leibniz (1686): Introduced concept of vis viva (living force), precursor to kinetic energy
  • Joseph-Louis Lagrange (1788): Developed analytical mechanics using potential energy
  • William Rowan Hamilton (1830s): Formulated Hamiltonian mechanics unifying energy concepts
  • Hermann von Helmholtz (1847): Stated law of conservation of energy clearly

These developments established the fundamental principles of energy conservation that underpin modern physics.

Introduction to Conservative Forces

🔬 What are Conservative Forces?

In physics, we encounter two fundamental types of forces:

  • Conservative Forces: Work done depends only on initial and final positions, not the path taken
  • Non-Conservative Forces: Work done depends on the path taken between points

Conservative forces are characterized by their ability to store energy in the configuration of a system, which we call potential energy.

📝 Formal Definitions

Definition 1: A force is conservative if the work done by the force on a body depends only on the initial and final positions and is independent of the path taken between these two points.

Definition 2: A force is conservative if the net work done by the force on a body moving along any closed path is zero.

These two definitions are mathematically equivalent and provide complementary ways to identify conservative forces.

📊 Conservative vs Non-Conservative Forces

Feature Conservative Forces Non-Conservative Forces
Work along closed path Zero Non-zero
Path dependence Independent Dependent
Potential energy Can be defined Cannot be defined
Energy transformation Reversible Irreversible
Examples Gravity, Spring force, Electrostatic Friction, Air resistance, Viscous drag

Spring Force as Conservative Force

🔬 Spring Force Characteristics

The spring force is a classic example of a conservative force. According to Hooke's Law, the force exerted by a spring is:

\[ F(x) = -kx \]

where k is the spring constant and x is the displacement from equilibrium position.

Work Done in Complete Cycle

🧪 Spring Force Demonstration

Consider a block of mass m attached to a spring with spring constant k, moving on a horizontal frictionless table.

Figure 1: Spring System

[Diagram: Mass-spring system showing displacement from equilibrium]

Setup: External agent displaces object from mean position (x = 0) to extreme position (x = +A)

Process: External agent is removed at t = 0, spring begins to do work on block

Work Calculation in Complete Cycle

Motion from x = +A to x = 0:

\[ W_1 = \frac{1}{2} kA^2 \]

Spring does positive work on block

Motion from x = 0 to x = -A:

\[ W_2 = -\frac{1}{2} kA^2 \]

Spring force reverses direction, does negative work

Motion from x = -A to x = 0:

\[ W_3 = \frac{1}{2} kA^2 \]

Spring does positive work again

Motion from x = 0 to x = +A:

\[ W_4 = -\frac{1}{2} kA^2 \]

Spring does negative work as block slows down

Total work in complete cycle:

\[ W_r = W_1 + W_2 + W_3 + W_4 \]
\[ = \frac{1}{2}kA^2 + \left( -\frac{1}{2}kA^2 \right) + \frac{1}{2}kA^2 + \left( -\frac{1}{2}kA^2 \right) \]
\[ = 0 \]

Conclusion: Since the net work done by the spring force over a complete cycle is zero, the spring force is a conservative force.

Path Independence

Path Independence Demonstration

Consider a mass-spring system where the block moves from x = +A to x = -A/2 along two different paths. Show that work done is the same for both paths.

Path 1: Direct Path

Block moves directly from x = +A to x = -A/2

\[ W_1 = \int_{+A}^{-A/2} (-kx) dx \]
\[ = \left[ -\frac{1}{2}kx^2 \right]_{+A}^{-A/2} \]
\[ = -\frac{1}{2}k \left[ \left( -\frac{A}{2} \right)^2 - A^2 \right] \]
\[ = -\frac{1}{2}k \left[ \frac{A^2}{4} - A^2 \right] \]
\[ = -\frac{1}{2}k \left[ -\frac{3A^2}{4} \right] \]
\[ = \frac{3}{8}kA^2 \]

Path 2: Via Equilibrium

Block moves from x = +A to x = 0, then from x = 0 to x = -A/2

\[ W_2 = \int_{+A}^{0} (-kx) dx + \int_{0}^{-A/2} (-kx) dx \]
\[ = \left[ -\frac{1}{2}kx^2 \right]_{+A}^{0} + \left[ -\frac{1}{2}kx^2 \right]_{0}^{-A/2} \]
\[ = \left[ 0 - \left( -\frac{1}{2}kA^2 \right) \right] + \left[ -\frac{1}{2}k \left( \frac{A^2}{4} \right) - 0 \right] \]
\[ = \frac{1}{2}kA^2 - \frac{1}{8}kA^2 \]
\[ = \frac{3}{8}kA^2 \]

Conclusion: Both paths give the same work done, confirming that spring force is path-independent and therefore conservative.

Gravitational Force as Conservative Force

🔬 Gravitational Force Characteristics

Near Earth's surface, the gravitational force on a mass m is approximately constant:

\[ \vec{F_g} = -mg\hat{j} \]

where g is the acceleration due to gravity and the negative sign indicates downward direction.

🧪 Gravitational Force Demonstration

Figure 2: Gravitational Paths

[Diagram: Three different paths between points A and B under gravity]

Setup: Consider three different paths between points A and B at different heights

Observation: Work done by gravity depends only on vertical displacement, not the path taken

Work Calculation for Different Paths

Path 1: Direct vertical path

\[ W_1 = \int_{y_1}^{y_2} (-mg) dy \]
\[ = -mg(y_2 - y_1) \]

Path 2: Inclined straight path

\[ W_2 = \int \vec{F} \cdot d\vec{s} \]
\[ = \int (-mg\hat{j}) \cdot (dx\hat{i} + dy\hat{j}) \]
\[ = \int -mg dy \]
\[ = -mg(y_2 - y_1) \]

Path 3: Arbitrary curved path

\[ W_3 = \int \vec{F} \cdot d\vec{s} \]
\[ = \int (-mg\hat{j}) \cdot (dx\hat{i} + dy\hat{j}) \]
\[ = \int -mg dy \]
\[ = -mg(y_2 - y_1) \]

Conclusion: All paths give the same work done: \( W = -mg(y_2 - y_1) \), confirming that gravitational force is conservative.

Potential Energy

🔬 Potential Energy Definition

For conservative forces, we can define a potential energy function U such that:

\[ W_{AB} = U_A - U_B = -\Delta U \]

where \( W_{AB} \) is the work done by the conservative force in moving from point A to point B.

The negative sign indicates that when a conservative force does positive work, the potential energy decreases.

Definition and Relation to Work

🧮 Potential Energy Derivation

Step 1: Work-Energy Relation

\[ W_{AB} = K_B - K_A \]

where K is kinetic energy

Step 2: For Conservative Forces

\[ W_{AB} = U_A - U_B \]

by definition of potential energy

Step 3: Equating Both Expressions

\[ K_B - K_A = U_A - U_B \]
\[ K_A + U_A = K_B + U_B \]

Step 4: Conservation of Mechanical Energy

\[ E = K + U = \text{constant} \]

for systems with only conservative forces

Conservation of Mechanical Energy

📝 Law of Conservation of Mechanical Energy

When only conservative forces act on a system, the total mechanical energy (sum of kinetic and potential energies) remains constant:

\[ K + U = \text{constant} \]

This is one of the most fundamental and powerful principles in physics, allowing us to solve complex problems without detailed knowledge of the forces involved.

🚀 Projectile Motion

In projectile motion (neglecting air resistance), mechanical energy is conserved. The sum of kinetic and gravitational potential energy remains constant throughout the trajectory.

🎢 Roller Coaster Physics

Roller coasters demonstrate energy conservation beautifully. The total mechanical energy (kinetic + gravitational potential) remains nearly constant, explaining why the coaster can't rise higher than its starting point.

🔬 Spring-Mass Systems

In ideal spring-mass systems, energy continuously transforms between kinetic energy and spring potential energy, with the total remaining constant.

One-Dimensional Conservative Systems

🔬 Force from Potential Energy

In one-dimensional systems, the relationship between force and potential energy is:

\[ F(x) = -\frac{dU}{dx} \]

The negative sign indicates that force points in the direction of decreasing potential energy.

Force from Potential Energy

🧮 Force-Potential Relationship Derivation

Step 1: Work in Small Displacement

\[ dW = F(x) dx \]

Step 2: Work and Potential Energy

\[ dW = -dU \]

Step 3: Equating Both Expressions

\[ F(x) dx = -dU \]
\[ F(x) = -\frac{dU}{dx} \]

Spring Potential Energy

🧮 Spring Potential Energy Derivation

Step 1: Spring Force

\[ F(x) = -kx \]

Step 2: Force-Potential Relationship

\[ F(x) = -\frac{dU}{dx} \]
\[ -\frac{dU}{dx} = -kx \]
\[ \frac{dU}{dx} = kx \]

Step 3: Integration

\[ U(x) = \int kx dx \]
\[ = \frac{1}{2}kx^2 + C \]

Step 4: Choosing Reference Point

Choosing U(0) = 0 (potential energy zero at equilibrium):

\[ U(x) = \frac{1}{2}kx^2 \]

Gravitational Potential Energy

🧮 Gravitational Potential Energy Derivation

Step 1: Gravitational Force

\[ F(y) = -mg \]

Step 2: Force-Potential Relationship

\[ F(y) = -\frac{dU}{dy} \]
\[ -\frac{dU}{dy} = -mg \]
\[ \frac{dU}{dy} = mg \]

Step 3: Integration

\[ U(y) = \int mg dy \]
\[ = mgy + C \]

Step 4: Choosing Reference Point

Choosing U(0) = 0 (potential energy zero at y = 0):

\[ U(y) = mgy \]

Equilibrium Points and Stability

🔬 Equilibrium Points

In a conservative system, equilibrium points occur where the net force is zero:

\[ F(x) = -\frac{dU}{dx} = 0 \]

This corresponds to critical points (maxima, minima, or inflection points) of the potential energy function.

Stable and Unstable Equilibrium

📊 Types of Equilibrium

Type Condition Force Behavior Energy Behavior
Stable Equilibrium \(\frac{d^2U}{dx^2} > 0\) Restoring force towards equilibrium Local minimum of potential energy
Unstable Equilibrium \(\frac{d^2U}{dx^2} < 0\) Force away from equilibrium Local maximum of potential energy
Neutral Equilibrium \(\frac{d^2U}{dx^2} = 0\) No net force Constant potential energy

Energy Method Analysis

Energy Method Example

Analyze the equilibrium points and stability for a particle moving in the potential \( U(x) = ax^2 - bx^4 \), where a and b are positive constants.

Step 1: Find Equilibrium Points

\[ F(x) = -\frac{dU}{dx} = -\frac{d}{dx}(ax^2 - bx^4) \]
\[ = -(2ax - 4bx^3) \]
\[ = -2ax + 4bx^3 \]

Set F(x) = 0 for equilibrium:

\[ -2ax + 4bx^3 = 0 \]
\[ 2x(-a + 2bx^2) = 0 \]
\[ x = 0 \quad \text{or} \quad x = \pm\sqrt{\frac{a}{2b}} \]

Step 2: Analyze Stability

\[ \frac{d^2U}{dx^2} = \frac{d}{dx}(2ax - 4bx^3) \]
\[ = 2a - 12bx^2 \]

At x = 0:

\[ \frac{d^2U}{dx^2} = 2a > 0 \]

⇒ Stable equilibrium (local minimum)

At \( x = \pm\sqrt{\frac{a}{2b}} \):

\[ \frac{d^2U}{dx^2} = 2a - 12b\left(\frac{a}{2b}\right) \]
\[ = 2a - 6a = -4a < 0 \]

⇒ Unstable equilibrium (local maxima)

Diatomic Molecule Problem

Diatomic Molecule Potential

The potential energy function for a diatomic molecule can be approximated by the Lennard-Jones potential:

\[ U(r) = 4\epsilon\left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right] \]

where r is the interatomic distance, ε is the depth of the potential well, and σ is the finite distance where U(r) = 0.

Step 1: Find Equilibrium Separation

\[ F(r) = -\frac{dU}{dr} \]
\[ = -4\epsilon\left[ -12\sigma^{12}r^{-13} + 6\sigma^6r^{-7} \right] \]
\[ = 48\epsilon\sigma^{12}r^{-13} - 24\epsilon\sigma^6r^{-7} \]

Set F(r) = 0 for equilibrium:

\[ 48\epsilon\sigma^{12}r^{-13} - 24\epsilon\sigma^6r^{-7} = 0 \]
\[ 48\sigma^{12}r^{-13} = 24\sigma^6r^{-7} \]
\[ 2\sigma^6 = r^6 \]
\[ r = 2^{1/6}\sigma \]

Step 2: Verify Stability

\[ \frac{d^2U}{dr^2} = \frac{d}{dr}(-48\epsilon\sigma^{12}r^{-13} + 24\epsilon\sigma^6r^{-7}) \]
\[ = 624\epsilon\sigma^{12}r^{-14} - 168\epsilon\sigma^6r^{-8} \]

At \( r = 2^{1/6}\sigma \):

\[ \frac{d^2U}{dr^2} = 624\epsilon\sigma^{12}(2^{1/6}\sigma)^{-14} - 168\epsilon\sigma^6(2^{1/6}\sigma)^{-8} \]
\[ = 624\epsilon\sigma^{-2}2^{-14/6} - 168\epsilon\sigma^{-2}2^{-8/6} \]
\[ = \frac{\epsilon}{\sigma^2}(624\cdot 2^{-7/3} - 168\cdot 2^{-4/3}) \]
\[ > 0 \quad \text{(positive, hence stable)} \]

Conclusion: The diatomic molecule has a stable equilibrium at \( r = 2^{1/6}\sigma \), which represents the bond length where the molecule is most stable.

Multi-Dimensional Conservative Systems

🔬 Conservative Forces in Multiple Dimensions

In three dimensions, a force is conservative if it can be written as the negative gradient of a scalar potential:

\[ \vec{F} = -\vec{\nabla}U = -\left( \frac{\partial U}{\partial x}\hat{i} + \frac{\partial U}{\partial y}\hat{j} + \frac{\partial U}{\partial z}\hat{k} \right) \]

This is equivalent to the condition that the curl of the force field is zero: \( \vec{\nabla} \times \vec{F} = 0 \).

📝 Path Independence in Multiple Dimensions

For a force to be conservative in three dimensions, the work done must be path-independent, which is equivalent to any of these conditions:

  1. \( \vec{\nabla} \times \vec{F} = 0 \) (curl is zero everywhere)
  2. \( \oint_C \vec{F} \cdot d\vec{r} = 0 \) for any closed path C
  3. \( \vec{F} = -\vec{\nabla}U \) for some scalar function U

These conditions ensure that the force field is "irrotational" and that energy is conserved.

Frequently Asked Questions

❓ Why is friction a non-conservative force?

Friction is non-conservative because:

  • Path Dependence: The work done by friction depends on the path length - longer paths dissipate more energy
  • Non-zero Closed Path Work: For any closed path where an object returns to its starting point, friction always does negative work
  • Energy Dissipation: Friction converts mechanical energy into thermal energy, which cannot be fully recovered
  • No Potential Energy: We cannot define a potential energy function for friction

This is why systems with friction don't conserve mechanical energy - some energy is always lost as heat.

❓ Can magnetic forces be conservative?

Magnetic forces present an interesting case:

  • Work Done: Magnetic forces do no work on charged particles because the force is always perpendicular to velocity (\( \vec{F} \cdot d\vec{r} = 0 \))
  • Path Dependence: While the work is zero for any path, magnetic forces can change the direction of motion without changing speed
  • Potential Energy: We cannot define a scalar potential energy for magnetic forces in the same way as for conservative forces
  • Vector Potential: In advanced electromagnetism, magnetic fields are described using a vector potential rather than a scalar potential

So while magnetic forces don't fit neatly into the conservative/non-conservative classification, they don't change the mechanical energy of systems.

❓ How do we identify conservative forces in real-world problems?

Here's a practical approach to identify conservative forces:

  • Check for Path Independence: If work depends only on endpoints, the force is conservative
  • Look for Energy Storage: Forces that can store energy (springs, gravity) are typically conservative
  • Identify Reversible Processes: If the force allows reversible energy transformation, it's likely conservative
  • Mathematical Test: In 3D, check if \( \vec{\nabla} \times \vec{F} = 0 \)
  • Common Conservative Forces: Gravity, spring forces, electrostatic forces
  • Common Non-Conservative Forces: Friction, air resistance, viscous drag, tension in ropes with friction

When solving problems, first identify which forces are conservative, as this allows using energy conservation methods.

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