Quick Calculator — Gravitational Potential Energy
Enter mass, height and (optional) gravity to compute potential energy in joules.
Formula: PE = m × g × h Units: mass in kg, g in m/s², height in m → Result in Joules (J)
Try these example scenarios
- Example A: A 2 kg object raised 5 m above the ground (use default g). Expected PE = 98.07 J.
- Example B: A 500 g book on a 1.2 m shelf. Convert mass to kg (0.5 kg) → PE ≈ 5.884 J.
- Example C: A 70 kg person standing on a 2 m platform → PE ≈ 1372.9 J.
What the calculator does
This calculator converts units you choose (grams → kg, cm → m, pounds → kg, feet → m) then computes the gravitational potential energy using PE = m·g·h. It also converts the energy into kJ or calories if requested.
In-depth Explanation & Derivation
The concept of potential energy comes from the work required to move an object against a force. For gravity near Earth’s surface, the force acting on a mass m is its weight: F = m × g. The work done to lift that mass through a height h is the force times distance (assuming constant g):
Work = Force × distance = (m × g) × h = m × g × h
This work is stored as gravitational potential energy. The unit of energy is the Joule (J), which is equivalent to N·m (Newton-meter). 1 Joule = 1 kg·m²/s².
When is this formula valid?
- On or near Earth’s surface where g is approximately constant.
- For objects where the height change is small relative to Earth’s radius.
- When gravitational field is uniform over the displacement.
Limitations & extensions
For very large height changes (e.g., satellites), use the general gravitational potential energy formula: U = -G·M·m / r, where G is the gravitational constant and r is the distance between centers.
Conservation of Energy (brief)
In a closed system, energy converts between kinetic and potential forms. If an object falls from height h with initial PE and no friction, its potential energy converts to kinetic energy (KE):
m·g·h = 1/2·m·v² → v = sqrt(2·g·h)
Worked Examples (Step-by-step)
Example 1 — Small object
Problem: A 250 g mass is lifted to a shelf 1.6 m above the floor. Calculate the potential energy.
Step 1: Convert mass to kg: 250 g = 0.250 kg.
Step 2: Use g = 9.80665 m/s² → PE = 0.250 × 9.80665 × 1.6
Calculation: PE ≈ 0.250 × 9.80665 × 1.6 = 3.92266 J (≈3.923 J)
Example 2 — Human on platform
Problem: A 68 kg person stands on a 3 m balcony. What is their gravitational potential energy relative to the ground?
Calculation: PE = 68 × 9.80665 × 3 = 2000.959 J ≈ 2.001 kJ
Example 3 — Energy in hydro applications (illustrative)
If 1000 kg of water is held at a height of 10 m, PE = 1000 × 9.80665 × 10 ≈ 98,066.5 J (~98.07 kJ). Hydroelectric plants convert such potential energy to electrical energy (with losses).
Units, Conversions & Practical Notes
Always keep units consistent. Mass should be in kilograms and height in meters for the standard SI result (Joules). If you prefer imperial units, convert them first.
- Mass: g → kg (divide by 1000); lb → kg (multiply by 0.45359237)
- Height: cm → m (divide by 100); ft → m (multiply by 0.3048)
- Energy: J → kJ (divide by 1000); J → cal (multiply by 0.239005736)
Tip: If you need to account for buoyancy, fluid lift, or non-uniform fields, use more advanced formulas. This calculator focuses on gravitational potential energy in a uniform field.
Frequently Asked Questions (FAQs)
Q1: Why must mass be in kilograms?
A: The SI unit for mass is kilogram. The Joule (SI unit of energy) is derived using kg, m and s (kg·m²/s²). If you input grams or pounds, convert them to kg first (the calculator does this automatically when you select units).
Q2: Can I use a different gravity value?
A: Yes. On other planets or in high-precision problems, use the local acceleration due to gravity (e.g., Moon g ≈ 1.62 m/s²).
Q3: Does potential energy depend on reference point?
A: Potential energy is relative; only differences in potential energy are physically meaningful. Choose a convenient zero level (ground, floor, etc.).
Q4: Is rotational potential energy included?
A: No. Rotational energy and elastic potential energy are different forms and require separate formulas.
Q5: Why do we ignore Earth's curvature?
A: For small height changes compared to Earth's radius (~6.37×10⁶ m), g is effectively constant and the simple m·g·h formula is accurate.