Potential Energy Calculator
Compute gravitational or spring potential energy, or solve for mass or height/displacement using standard physics equations.
Calculation Steps
Practical Guide
Potential Energy Calculator
Potential energy is the stored energy of position or configuration. This guide explains the two most common forms—gravitational and elastic (spring)—and shows you how to use the calculator to solve for energy, mass, or height/displacement. You’ll see the underlying equations, typical assumptions, worked examples, and sanity checks so your answers make physical sense in real projects.
Quick Start
The calculator lets you solve for potential energy \(U\) or rearrange the equation to solve for a missing input. Follow these steps to avoid the most common mistakes.
- 1Choose the Potential Energy Type: Gravitational \((U=mgh)\) or Elastic/Spring \((U=\tfrac12 kx^2)\).
- 2Select what you want to Solve For: Potential Energy \(U\), Mass \(m\), or Height/Displacement \(h\) or \(x\). The solved variable’s row will hide automatically.
- 3Enter the known inputs. Use realistic values and confirm each unit selector matches what you typed.
- 4If solving for mass or height/displacement, enter the Given Potential Energy and its unit.
- 5For gravitational problems, verify gravity \(g\). The default is Earth average \(9.80665\,\text{m/s}^2\), but you can change it.
- 6Read the result and check Quick Stats to compare energy in several unit systems.
- 7Open Show Steps to confirm the equation path matches your scenario.
Tip: If you’re unsure which type applies, ask “Is energy stored because of height/position (gravitational) or because something is stretched/compressed (elastic)?”
Common pitfall: Make sure height/displacement is relative to a clear reference level. Potential energy is defined up to a constant, so you must pick a consistent zero.
Choosing Your Method
Potential energy shows up across disciplines—from crane lifts and water tanks to vehicle suspensions and lab springs. The calculator supports two main methods and their rearrangements.
Method A — Gravitational Potential Energy
Use this when energy is stored because a mass is elevated in a gravitational field. The core equation is:
- Best for lifting, dropping, reservoir head, hydro systems, hoists, and slopes.
- Linear in mass and height, so sensitivity is easy to interpret.
- Good approximation near Earth’s surface where \(g\) is constant.
- Assumes uniform gravity and small height change relative to Earth’s radius.
- Ignores rotational or relativistic effects in extreme cases.
Method B — Elastic (Spring) Potential Energy
Use this when energy is stored by stretching or compressing an elastic element that behaves linearly. The core equation is:
- Best for springs, elastic cables, energy absorbers, and small-strain materials.
- Captures the nonlinear increase in stored energy with displacement.
- Pairs naturally with Hooke’s law \(F = kx\).
- Requires the linear-elastic assumption; real materials may yield or stiffen.
- Only valid within the spring’s rated displacement range.
Method C — Solve for Missing Variable
In design you often know the required stored energy and need a missing input. The calculator rearranges equations automatically:
- Great for sizing: “How heavy can I lift to a given height?” or “What displacement stores a target energy?”
- Prevents algebra mistakes and handles unit conversions cleanly.
- Still depends on correct scenario selection and input realism.
What Moves the Number the Most
Potential energy is sensitive to a small set of dominant variables. Understanding these “levers” helps you interpret the output and spot errors immediately.
Energy scales linearly with mass. Doubling mass doubles \(U\). If your result seems high, check whether you accidentally entered pounds while the unit selector is set to kilograms.
Also linear. A 2× increase in height gives 2× energy, assuming \(g\) constant. The reference level matters: \(h\) is the change in elevation from your zero datum.
Usually fixed at Earth average, but local variation (latitude, altitude) changes \(g\) by <1%. It matters more off-planet or in very tall-structure analyses.
Higher stiffness stores more energy for the same displacement. Because \(U\propto k\), a 20% increase in \(k\) yields a 20% increase in energy at fixed \(x\).
The big driver in elastic energy. Since \(U\propto x^2\), doubling \(x\) quadruples energy. Small measurement errors in \(x\) can cause large changes in \(U\).
The equation assumes Hookean behavior. If the spring or material is near yield, the real energy may be lower (softening) or higher (hardening) than the calculator predicts.
Worked Examples
These examples mirror real engineering tasks. Follow the same setup in the calculator to reproduce the results.
Example 1 — Gravitational Energy of a Lifted Load
- Scenario: A hoist lifts a steel crate.
- Mass: \(m = 250\,\text{kg}\).
- Height increase: \(h = 12\,\text{m}\).
- Gravity: \(g = 9.80665\,\text{m/s}^2\).
Calculator setup: Choose “Gravitational,” solve for “Potential Energy,” enter 250 kg and 12 m, keep \(g\) default. Output should read about 29.4 kJ.
Example 2 — Required Spring Displacement for a Target Energy
- Scenario: A safety bumper must absorb \(U = 500\,\text{J}\).
- Spring constant: \(k = 18\,000\,\text{N/m}\).
- Unknown: displacement \(x\).
Calculator setup: Choose “Elastic/Spring,” solve for “Height/Displacement,” enter Given Energy = 500 J and k = 18000 N/m. Result should be about 0.236 m.
Common Layouts & Variations
Potential energy equations appear in many configurations. The table below lists the most common uses and what assumptions are typically acceptable.
| Configuration / Use Case | Equation Form | Typical Assumptions | Engineering Notes |
|---|---|---|---|
| Lifted mass (cranes, elevators) | \(U=mgh\) | Constant \(g\), rigid body | Use change in elevation; add motor efficiency separately if estimating power. |
| Water reservoir head | \(U=\rho g V h\) | Incompressible fluid, uniform density | Convert volume to mass; reference height is to turbine datum or outlet. |
| Vehicle suspension compression | \(U=\tfrac12 kx^2\) | Linear spring rate near operating point | Real suspensions may be progressive; check manufacturer curves. |
| Elastic cable or rod in tension | \(U=\int_0^x F\,dx\) | Linear range \(\Rightarrow U=\tfrac12 kx^2\) | If stiffness varies, a constant-\(k\) simplification is only approximate. |
| Gravitational energy on slopes | \(U=mg\Delta z\) | Small angle changes OK | Use vertical rise \(\Delta z\), not path length. |
- Use vertical height change for gravitational cases.
- Confirm the spring is operating within rated linear travel.
- Track sign conventions separately; energy is always non-negative.
- Keep a consistent reference level for all heights in a system model.
- Convert volume to mass using density when needed.
- Don’t mix lbf and lbm without selecting the right units.
Specs, Logistics & Sanity Checks
Because potential energy is a simple concept, errors usually come from bad inputs or mismatched assumptions. Use these checks before you lock in a design or report.
Units to Verify
- Mass: kg vs lb (weight) are not interchangeable without conversion.
- Length: m, ft, in, mm—watch for mixed drawings and field notes.
- Spring rate: N/m vs lbf/in changes results by orders of magnitude.
- Energy: J, kJ, Wh, ft·lbf—choose what your audience expects.
Physical Reasonableness
- If a small displacement gives huge energy, recheck \(k\) units or magnitude.
- If energy is near zero for a tall lift, check that height wasn’t entered in cm or in while units say m or ft.
- Gravitational energy should scale directly with both mass and height.
When the Simple Model Breaks
- Very large heights (space/planetary): \(g\) varies with radius, so \(U=mGM(1/r)\) is more accurate.
- Nonlinear springs or plastic deformation: use the full force-displacement curve \(U=\int F\,dx\).
- Dynamic events: potential energy converts to kinetic; add damping and losses separately.
In practice, engineers use potential energy as one piece of a system energy balance. For example, if a lifted load is then lowered through friction, your total usable energy is less than \(mgh\). The calculator gives the ideal stored energy; you add efficiencies and loss terms as needed.