Potential Energy Calculator

Professional potential energy calculator for physics. Compute gravitational potential energy (m·g·h and −G·m1·m2/r) and elastic spring energy (½·k·x²) with precise unit conversions. Fully accessible, mobile‑first, and optimized for performance.

Potential Energy Calculator

This professional-grade tool computes potential energy for three common models: gravitational (near-surface), gravitational (two-body), and elastic (spring). It’s built for students, educators, and engineers who need precise results with transparent methodology, accessible UX, and robust unit conversions.

Calculator

Choose model
Height is measured from your chosen reference level. Negative heights are allowed and yield negative energy relative to that reference.
Standard gravity on Earth is g₀ = 9.80665 m/s² (ISO 80000-3). Edit if using a different location or a precise local value.

Results

Potential Energy (J) —
Potential Energy (kJ) —
Potential Energy (kWh) —

Data Source and Methodology

  • CODATA 2018 Recommended Values of the Fundamental Physical Constants (G = 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻²), National Institute of Standards and Technology (NIST), 2019. Link: https://physics.nist.gov/cuu/Constants/
  • ISO 80000-3:2019 Quantities and units — Part 3: Space and time (standard gravity g₀ = 9.80665 m/s²). Link: https://www.iso.org/standard/64972.html

All calculations are strictly based on the formulas and data provided by this source.

The Formula Explained

Near-surface gravity: \( U = m \cdot g \cdot h \)
Two-body gravity: \( U = -\,\dfrac{G\,m_1\,m_2}{r} \)
Ideal spring: \( U = \tfrac{1}{2}\,k\,x^2 \)

Internally, all inputs are converted to SI units before evaluation to ensure numerical consistency and precision.

Glossary of Variables

  • U — Potential energy (result), in joules [J].
  • m, m₁, m₂ — Mass, in kilograms [kg].
  • h — Height relative to a reference, in meters [m].
  • g — Gravitational acceleration, in meters per second squared [m/s²].
  • G — Universal gravitational constant, 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻² (CODATA 2018).
  • r — Separation between two centers of mass, in meters [m].
  • k — Spring constant, in newtons per meter [N/m].
  • x — Spring displacement from equilibrium, in meters [m].

How It Works: A Step‑by‑Step Example

Scenario: A 2 kg backpack is lifted to a shelf 1.8 m high on Earth.

  1. Choose the model “Gravitational (near-surface)”.
  2. Enter: m = 2 kg, h = 1.8 m, g = 9.80665 m/s².
  3. Compute using \( U = m \cdot g \cdot h \).
  4. Result: \( U = 2 \times 9.80665 \times 1.8 \approx 35.3 \text{ J} \) (≈ 0.0353 kJ).

Tip: For very large altitude changes (e.g., satellite orbits), use the two-body formula \( U = -\dfrac{G m_1 m_2}{r} \) for higher fidelity.

Frequently Asked Questions (FAQ)

What is potential energy?

Potential energy is stored energy due to position or configuration—such as height in a gravitational field or compression/extension of a spring.

Which formula should I use?

Use m·g·h for small height differences near a planet’s surface (constant g). Use −G·m1·m2/r for large distances or celestial mechanics. Use ½·k·x² for ideal springs.

Can the calculator handle negative heights?

Yes. Height is relative to a reference level you choose. Negative height yields negative potential energy relative to that reference.

What unit systems are supported?

Mass: kg, g, lb; Length: m, cm, mm, ft, in, km, mi; Spring constant: N/m, N/mm, lb/in, lb/ft. All inputs are converted to SI for computation.

Why is two-body gravitational energy negative?

By convention, potential energy is zero at infinite separation. Bringing masses together reduces potential energy, so values are negative.

Is air resistance considered?

No. Potential energy is a state function independent of the path taken. Dissipative effects like air drag affect kinetic energy, not potential energy itself.


Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[','\]
','
Formula (extracted text)
Near-surface gravity: \( U = m \cdot g \cdot h \) Two-body gravity: \( U = -\,\dfrac{G\,m_1\,m_2}{r} \) Ideal spring: \( U = \tfrac{1}{2}\,k\,x^2 \)
Variables and units
  • No variables provided in audit spec.
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

Full original guide (expanded)

Potential Energy Calculator

This professional-grade tool computes potential energy for three common models: gravitational (near-surface), gravitational (two-body), and elastic (spring). It’s built for students, educators, and engineers who need precise results with transparent methodology, accessible UX, and robust unit conversions.

Calculator

Choose model
Height is measured from your chosen reference level. Negative heights are allowed and yield negative energy relative to that reference.
Standard gravity on Earth is g₀ = 9.80665 m/s² (ISO 80000-3). Edit if using a different location or a precise local value.

Results

Potential Energy (J) —
Potential Energy (kJ) —
Potential Energy (kWh) —

Data Source and Methodology

  • CODATA 2018 Recommended Values of the Fundamental Physical Constants (G = 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻²), National Institute of Standards and Technology (NIST), 2019. Link: https://physics.nist.gov/cuu/Constants/
  • ISO 80000-3:2019 Quantities and units — Part 3: Space and time (standard gravity g₀ = 9.80665 m/s²). Link: https://www.iso.org/standard/64972.html

All calculations are strictly based on the formulas and data provided by this source.

The Formula Explained

Near-surface gravity: \( U = m \cdot g \cdot h \)
Two-body gravity: \( U = -\,\dfrac{G\,m_1\,m_2}{r} \)
Ideal spring: \( U = \tfrac{1}{2}\,k\,x^2 \)

Internally, all inputs are converted to SI units before evaluation to ensure numerical consistency and precision.

Glossary of Variables

  • U — Potential energy (result), in joules [J].
  • m, m₁, m₂ — Mass, in kilograms [kg].
  • h — Height relative to a reference, in meters [m].
  • g — Gravitational acceleration, in meters per second squared [m/s²].
  • G — Universal gravitational constant, 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻² (CODATA 2018).
  • r — Separation between two centers of mass, in meters [m].
  • k — Spring constant, in newtons per meter [N/m].
  • x — Spring displacement from equilibrium, in meters [m].

How It Works: A Step‑by‑Step Example

Scenario: A 2 kg backpack is lifted to a shelf 1.8 m high on Earth.

  1. Choose the model “Gravitational (near-surface)”.
  2. Enter: m = 2 kg, h = 1.8 m, g = 9.80665 m/s².
  3. Compute using \( U = m \cdot g \cdot h \).
  4. Result: \( U = 2 \times 9.80665 \times 1.8 \approx 35.3 \text{ J} \) (≈ 0.0353 kJ).

Tip: For very large altitude changes (e.g., satellite orbits), use the two-body formula \( U = -\dfrac{G m_1 m_2}{r} \) for higher fidelity.

Frequently Asked Questions (FAQ)

What is potential energy?

Potential energy is stored energy due to position or configuration—such as height in a gravitational field or compression/extension of a spring.

Which formula should I use?

Use m·g·h for small height differences near a planet’s surface (constant g). Use −G·m1·m2/r for large distances or celestial mechanics. Use ½·k·x² for ideal springs.

Can the calculator handle negative heights?

Yes. Height is relative to a reference level you choose. Negative height yields negative potential energy relative to that reference.

What unit systems are supported?

Mass: kg, g, lb; Length: m, cm, mm, ft, in, km, mi; Spring constant: N/m, N/mm, lb/in, lb/ft. All inputs are converted to SI for computation.

Why is two-body gravitational energy negative?

By convention, potential energy is zero at infinite separation. Bringing masses together reduces potential energy, so values are negative.

Is air resistance considered?

No. Potential energy is a state function independent of the path taken. Dissipative effects like air drag affect kinetic energy, not potential energy itself.


Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[','\]
','
Formula (extracted text)
Near-surface gravity: \( U = m \cdot g \cdot h \) Two-body gravity: \( U = -\,\dfrac{G\,m_1\,m_2}{r} \) Ideal spring: \( U = \tfrac{1}{2}\,k\,x^2 \)
Variables and units
  • No variables provided in audit spec.
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

Potential Energy Calculator

This professional-grade tool computes potential energy for three common models: gravitational (near-surface), gravitational (two-body), and elastic (spring). It’s built for students, educators, and engineers who need precise results with transparent methodology, accessible UX, and robust unit conversions.

Calculator

Choose model
Height is measured from your chosen reference level. Negative heights are allowed and yield negative energy relative to that reference.
Standard gravity on Earth is g₀ = 9.80665 m/s² (ISO 80000-3). Edit if using a different location or a precise local value.

Results

Potential Energy (J) —
Potential Energy (kJ) —
Potential Energy (kWh) —

Data Source and Methodology

  • CODATA 2018 Recommended Values of the Fundamental Physical Constants (G = 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻²), National Institute of Standards and Technology (NIST), 2019. Link: https://physics.nist.gov/cuu/Constants/
  • ISO 80000-3:2019 Quantities and units — Part 3: Space and time (standard gravity g₀ = 9.80665 m/s²). Link: https://www.iso.org/standard/64972.html

All calculations are strictly based on the formulas and data provided by this source.

The Formula Explained

Near-surface gravity: \( U = m \cdot g \cdot h \)
Two-body gravity: \( U = -\,\dfrac{G\,m_1\,m_2}{r} \)
Ideal spring: \( U = \tfrac{1}{2}\,k\,x^2 \)

Internally, all inputs are converted to SI units before evaluation to ensure numerical consistency and precision.

Glossary of Variables

  • U — Potential energy (result), in joules [J].
  • m, m₁, m₂ — Mass, in kilograms [kg].
  • h — Height relative to a reference, in meters [m].
  • g — Gravitational acceleration, in meters per second squared [m/s²].
  • G — Universal gravitational constant, 6.67430×10⁻¹¹ m³·kg⁻¹·s⁻² (CODATA 2018).
  • r — Separation between two centers of mass, in meters [m].
  • k — Spring constant, in newtons per meter [N/m].
  • x — Spring displacement from equilibrium, in meters [m].

How It Works: A Step‑by‑Step Example

Scenario: A 2 kg backpack is lifted to a shelf 1.8 m high on Earth.

  1. Choose the model “Gravitational (near-surface)”.
  2. Enter: m = 2 kg, h = 1.8 m, g = 9.80665 m/s².
  3. Compute using \( U = m \cdot g \cdot h \).
  4. Result: \( U = 2 \times 9.80665 \times 1.8 \approx 35.3 \text{ J} \) (≈ 0.0353 kJ).

Tip: For very large altitude changes (e.g., satellite orbits), use the two-body formula \( U = -\dfrac{G m_1 m_2}{r} \) for higher fidelity.

Frequently Asked Questions (FAQ)

What is potential energy?

Potential energy is stored energy due to position or configuration—such as height in a gravitational field or compression/extension of a spring.

Which formula should I use?

Use m·g·h for small height differences near a planet’s surface (constant g). Use −G·m1·m2/r for large distances or celestial mechanics. Use ½·k·x² for ideal springs.

Can the calculator handle negative heights?

Yes. Height is relative to a reference level you choose. Negative height yields negative potential energy relative to that reference.

What unit systems are supported?

Mass: kg, g, lb; Length: m, cm, mm, ft, in, km, mi; Spring constant: N/m, N/mm, lb/in, lb/ft. All inputs are converted to SI for computation.

Why is two-body gravitational energy negative?

By convention, potential energy is zero at infinite separation. Bringing masses together reduces potential energy, so values are negative.

Is air resistance considered?

No. Potential energy is a state function independent of the path taken. Dissipative effects like air drag affect kinetic energy, not potential energy itself.


Audit: Complete
Formula (LaTeX) + variables + units
This section shows the formulas used by the calculator engine, plus variable definitions and units.
Formula (extracted LaTeX)
\[','\]
','
Formula (extracted text)
Near-surface gravity: \( U = m \cdot g \cdot h \) Two-body gravity: \( U = -\,\dfrac{G\,m_1\,m_2}{r} \) Ideal spring: \( U = \tfrac{1}{2}\,k\,x^2 \)
Variables and units
  • No variables provided in audit spec.
Sources (authoritative):
Changelog
Version: 0.1.0-draft
Last code update: 2026-01-19
0.1.0-draft · 2026-01-19
  • Initial audit spec draft generated from HTML extraction (review required).
  • Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
  • Confirm sources are authoritative and relevant to the calculator methodology.
Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn
Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft
Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog
  • 0.1.0-draft — 2026-01-19: Initial draft (review required).