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 → Gravitational force ↓


Gravitational potential energy


Gravitational potential energy is a fundamental concept in classical mechanics, particularly when discussing the workings of gravity, a force that affects all objects with mass. This concept derives from the potential energy stored within an object based on its distance from another object, usually a very large mass object such as the Earth. In this discussion, we will explore this concept in depth.

Basic definition

Gravitational potential energy (GPE) is the energy that an object has due to its position in a gravitational field. The classic formula for potential energy due to gravity is

U = mgh

Where:

  • U is the gravitational potential energy,
  • m is the mass of the object,
  • g is the acceleration due to gravity,
  • h is the height of the object above the reference point.

Understanding the formula

The formula U = mgh is straightforward when we consider its components:

  • Mass (m): The mass of the object is directly proportional to its GPE. If we double the mass, the potential energy will double.
  • Acceleration due to gravity (g): At the Earth's surface, g is about 9.81 m/s². This value can change slightly depending on location (altitude and latitude).
  • Altitude (h): This is the height of the object from the reference point (usually the ground). Increasing altitude increases GPE.

Visual example

Imagine a ball of mass m at the top of a hill of height h. GPE can be visualized in terms of the ball's ability to roll downhill under gravity.

H M

As the ball rolls down, the potential energy is converted into kinetic energy, which shows the principle of energy conservation. The height h changes, which directly affects the potential energy.

Reference point

Gravitational potential energy is a type of potential energy. It is important to understand that potential energy is defined relative to a point. In the above formula, the reference point of the GPE (where h = 0) is important. This point is often the ground, but it can be any level or location. This choice does not affect calculations within the same system, but it must be consistent to prevent errors.

Examples in everyday life

Let us illustrate this with some examples of gravitational potential energy:

  • Water in the reservoir: The water stored in the dam at a height has sufficient gravitational energy. When it is allowed to flow downhill, this energy can be converted into kinetic energy, which is then converted into electrical energy through a turbine in a hydroelectric power plant.
  • Climbing stairs: When you climb stairs, you lift your body against gravity, which increases your gravitational potential energy. If you weigh 70 kg and you rise 2 meters, using g = 9.81 m/s², your change in GPE is:
    • ΔU = mgΔh = 70 * 9.81 * 2 = 1373.4 J (Joules)
  • Roller coaster: A roller coaster car gains potential energy as it is pulled to the top of a hill. As it descends, this energy is converted into kinetic energy, causing the coaster to accelerate forward and climb subsequent hills.

The math behind gravitational potential energy

A simplified form of U = mgh; this applies when dealing with a uniform gravitational field, such as near the Earth's surface. In more general cases, especially at large distances, the formula becomes:

U = -G * (M * m) / r

Where:

  • G is the gravitational constant, about 6.674 × 10 -11 N(m/kg) 2
  • M is the mass of the Earth or other massive body,
  • r is the distance between the centers of the two masses (mass m and mass M).

This formula is derived from the universal law of gravitation. It shows that potential energy is negative over distances because we consider infinity to be a point of zero potential energy. Thus, gravitational forces are attractive.

Conservation of mechanical energy

In an isolated system, the total mechanical energy — the sum of kinetic energy (T) and potential energy (U) — remains constant, which can be expressed as:

E = T + U = constant

When only gravity acts, energy changes form but is never destroyed. For an object falling from a height, the increase in kinetic energy will be equal to the decrease in gravitational potential energy.

Text illustration

Consider a pendulum. At its highest point, momentarily, it is at rest with maximum potential and no kinetic energy. As it swings downward, the GPE is converted into kinetic energy until at the lowest point, its velocity is maximum, and the GPE is minimum. As the pendulum swings backward, the kinetic energy is converted back into potential energy in one cycle.

Relation with escape velocity

Gravitational potential energy is also related to the concept of escape velocity—the minimum speed needed to break free from a gravitational field without any additional acceleration.

The potential energy equation at a critical distance indicates the work required to move an object from the surface to the point at infinity where the effect of gravity ceases. The escape velocity is obtained by equating the kinetic energy at the surface of a massive body with this potential energy:

1/2 * m * v 2 = G * (M * m) / R

Solving for the value of escape velocity v gives:

v = sqrt(2 * G * M / R)

Here, R is the radius from the center of the massive body to its surface. Note that the escape velocity is independent of the mass of the object being projected.

Closing thoughts

Gravitational potential energy is an essential part of understanding mechanics within a gravitational field. It enables us to measure potential work and energy changes, which are important for explaining a variety of natural phenomena, engineering applications, and celestial mechanics.

From everyday scenarios such as lifting loads to calculating the trajectories of space missions, gravitational potential energy provides the framework for analyzing situations where gravity plays a major role. The concepts we discussed are fundamentally applicable to students and professionals in physics, engineering, and related disciplines.

Whether considering the cascade of water falling from a waterfall, the energy required for planets to orbit the Sun, or building energy-efficient structures, understanding gravitational potential energy is important.


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Gravitational potential energy

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