Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Work and Energy ↓


Conservative and non-conservative forces


In classical mechanics, it is necessary to understand the concepts of work and energy, and an important part of this understanding involves distinguishing between conservative and non-conservative forces. Forces affect the motion and energy of objects, and knowing how different forces affect energy is important in solving many physical problems.

What are conservative forces?

A conservative force is one in which the work done by the force on an object depends only on the initial and final position of the object, no matter what path it takes. This means that if an object moves from point A to point B, the work done by the conservative force is the same, no matter what trajectory it takes.

One of the most famous examples of a conservative force is the force of gravity. When an object is raised and then brought back to its original height, the net work done by gravity is zero, since gravity depends only on vertical position changes.

Another example is the elastic force exerted by an ideal spring. The work done when the spring is stretched or compressed depends only on the amount of stretch or compression, not on how the change occurred.

Mathematical representation

For any conservative force, the work W done by the force can be mathematically described as:

W = U(A) - U(B)

where U(A) and U(B) represent the potential energy at points A and B, respectively.

Visual example: conservative forces


        
        
        
        A
        B
        Path 1
        
        
        Path 2

In the above figure, whether the object takes path 1 or path 2, the work done by the conservative force from A to B is the same.

Properties of conservative forces

  • The work done by a conservative force in a closed loop is zero.
  • Conservative forces have potential energy associated with them.
  • Total mechanical energy (kinetic + potential) is conserved only in a system of conservative forces.

What are non-conservative forces?

Non-conservative forces are those for which the work done depends on the path taken by the object. This means that non-conservative forces dissipate mechanical energy, often converting it into thermal energy or other forms.

Examples of non-conservative forces

Friction is the most common example of a non-conservative force. When you slide an object across a surface, the energy expended in overcoming friction is not stored as potential or useful kinetic energy but is converted into heat.

Air resistance is another non-conservative force. When an object moves through air, the air molecules exert a force opposite to the motion, and over time, this reduces the kinetic energy of the moving object, often converting it into heat.

In general terms, non-conservative forces cannot conserve mechanical energy within a closed system, making it challenging to use energy methods to completely predict motion without knowing additional factors such as path length.

Visual example: non-conservative force


        
        
        
        
        A
        B
        clash
        (non-opposition)

In the example above, going from A to B with friction takes different work and energy than going back, because energy is dissipated as heat.

Properties of non-conservative forces

  • The work done is path dependent.
  • These have no associated potential energy.
  • They can convert mechanical energy into other forms, such as thermal energy.

Implications of conservative and non-conservative forces

The distinction between these types of forces is fundamental in understanding many physical scenarios. For example, in an idealized system where only conservative forces act, it is simple to calculate the change in energy or predict motion using conservation of mechanical energy.

When non-conservative forces are involved, additional calculations or measurements are necessary to account for energy transfer and decay, which means more information is needed to accurately estimate the state of the object.

Example: sliding a box

Imagine you are sliding a box across a carpeted floor. To keep the box moving at a constant speed, work must be done continuously against friction. This work goes into heat and sound, and is not recoverable in terms of potential or kinetic energy in the box.

Mathematical treatment of systems

In general, the work–energy principle is widely applied when analyzing a system under the influence of both conservative and non-conservative forces:

KE_initial + PE_initial + Work_non-conservative = KE_final + PE_final

Where KE denotes kinetic energy and PE denotes potential energy. The term Work_non-conservative accounts for the loss or gain by non-conservative forces.

Understanding path dependency

It is important to understand the path dependence of non-conservative forces versus the path independence of conservative forces in physics. This understanding helps determine whether energy conservation methods can be applied directly, or whether additional work calculations are necessary.

Consider a person using different paths to climb a hill. With gravity (conservative), the energy required depends only on the height difference, not on the path. With friction or resistance (non-conservative), the length and nature of the road matter significantly.

Summary

In conclusion, identifying and distinguishing between conservative and non-conservative forces helps us determine how energy is stored, transferred, or destroyed within a system. This understanding forms the basis for solving complex physical problems by allowing predictions in mechanical processes involving work and energy. Mastering these concepts is critical for progressing to more advanced topics in physics and engineering.


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Conservative and non-conservative forces

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