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 ↓


Potential energy in classical mechanics


In the study of classical mechanics, energy is a fundamental concept that helps us understand how objects interact with each other and their environment. Of the various forms of energy, potential energy is an important type that relates to the stored energy related to the position of an object or arrangement within a system.

Understanding energy

Before learning about potential energy in detail, let us briefly understand the concept of energy in physics. Energy is the potential to do work. Whether it is creating motion, generating heat or facilitating the flow of electricity, energy is a transferable and modifiable entity.

Work is the process of energy transfer into or out of a system, involving force and displacement. The formula for work is:

        Work (W) = Force (F) × Distance (d) × cos(θ)

Where:

  • F is the applied force,
  • d is the distance over which the force is applied,
  • θ is the angle between the force and the direction of motion.

Introduction to potential energy

Potential energy is energy stored in an object due to its position relative to other objects, tension within it, electric charge, or other factors. It is called "potential" because it has the potential to be converted into other energy forms, such as kinetic energy.

Gravitational potential energy

The most common type of potential energy is gravitational potential energy. This is the energy stored in an object when it is held above the ground. Gravitational potential energy is given by the formula:

        Potential Energy (PE) = m × g × h

Where:

  • m is the mass of the object (in kilograms),
  • g is the acceleration due to gravity (about 9.81 m/s2 at the Earth's surface),
  • h is the height of the object above the ground (in meters).

For example, when you lift a book off a table, you are doing work against the force of gravity, and the book gains potential energy. If you let go of the book, this potential energy is converted into kinetic energy as the book falls.

Book Height(H)

Elastic potential energy

Elastic potential energy is found in objects that can be stretched or compressed, such as springs, rubber bands, or elastic materials. Elastic potential energy is determined by how much the object is deformed and how stiff its material is. A general formula for calculating the potential energy stored in a spring is:

        Elastic Potential Energy (PE_elastic) = 1/2 × k × x^2

Where:

  • k is the spring constant, which is a measure of the stiffness of the spring,
  • x is the displacement of the spring from its equilibrium position (in meters).

Consider compressing a spring to a certain length; it stores potential energy. When it is released, the stored energy can be used to push or move objects.

Compressed Spring

Chemical potential energy

Chemical potential energy relates to the energy stored in the chemical bonds of molecules. This energy is observed when a chemical reaction occurs, breaking or forming new bonds, thereby releasing or absorbing energy. For example, food contains chemical potential energy. When eaten, our body metabolizes it, converting potential energy into kinetic energy to maintain activities and retain heat.

Potential energy in conservative forces

Potential energy is closely related to conservative forces, which are forces where the work done does not depend on the path taken but only on the initial and final conditions. Gravitational and elastic forces are examples of conservative forces. For these types of forces, the total mechanical energy in the system (the sum of potential and kinetic energy) remains constant if only conservative forces are acting.

Consider two scenarios:
1. A stone is hanging on a cliff and suddenly falls down.
2. The stone is sliding down a smooth, frictionless slope from the same height.
In both scenarios, the mechanical energy of the stone remains the same ignoring air resistance or other non-conservative forces, demonstrating energy conservation within conservative systems.

Visualization of potential energy

Imagine the roller coaster is at the top of its track. At the highest point, it has maximum gravitational potential energy due to its height. As it descends, this potential energy turns into kinetic energy as its speed increases.

Start braid Ending

Potential energy in daily life

Potential energy plays an important role in various daily activities. Consider the following examples:

  • Hydropower dams: Water stored at heights has gravitational potential energy. When released, it is converted into kinetic energy, which flows downhill to drive turbines and generate electricity.
  • Archery: When an archer draws a bow, work is done against the tension of the string, storing elastic potential energy in the bent bow. When released, this energy is transmitted to the arrow, propelling it forward at high speed.
  • Pneumatic systems: Using air pressure as the storage medium, systems such as air rifles store potential energy in compressed gas forms, and release the energy to perform work when triggered.

Synthesis of kinetic and potential energy

The relationship between kinetic and potential energy is important in many mechanical systems. As each form of energy is converted from one form to another, the total energy of the system remains constant, provided no external forces intervene. This dynamic exchange illustrates the guiding principle of energy conservation.

For example, the swing of a pendulum is a good example. At the highest point of its swing, the pendulum has the most potential energy and no kinetic energy. As it moves downward, the potential energy is converted into kinetic energy, reaching its highest level when it passes the lowest point of the swing.

Anchor

Concluding remarks

Potential energy is a silent observer in the countless interactions around us, shaping behavior and facilitating events in subtle but profound ways. From gravitational pull to the elasticity of matter and the silent strength of chemical bonds, potential energy remains an essential part of understanding the intricacies of the natural world.

The study of potential energy enables us to gain a deeper understanding of the trade-offs between energy storage and emissions, and drives innovation across a wide range of industries and scientific discoveries.


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Potential energy in classical mechanics

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