Potential Energy

Reference data and engineering information about potential energy for dynamics applications.

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Overview

Engineering reference data for Potential Energy in dynamics.

Key Formulas

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Work

$W = Fd\cos\theta$

Force × displacement × cos(angle).

Variables

Symbol	Description	Unit
$F$	Force	N
$m$	Mass	kg
$a$	Acceleration	m/s²
$v$	Velocity	m/s
$E_k$	Kinetic energy	J

Practical Examples

The fundamental relationship $E_p = m \cdot g \cdot \Delta h$ can be applied directly to engineering problems using either mass or force inputs.

SI Units Example: A mass of $1000 \, \text{kg}$ is elevated by $\Delta h = 10 \, \text{m}$ . $E_p = (1000 \, \text{kg}) \cdot (9.81 \, \text{m/s}^2) \cdot (10 \, \text{m}) = 98{,}100 \, \text{J} = 98 \, \text{kJ} \approx 0.027 \, \text{kWh}$

Imperial Units Example (Force): A body with a weight of $500 \, \text{lbf}$ is elevated by $\Delta h = 30 \, \text{ft}$ . $E_p = (500 \, \text{lbf}) \cdot (30 \, \text{ft}) = 15{,}000 \, \text{ft·lb}$

Imperial Units Example (Mass): A body with a mass of $15 \, \text{slugs}$ is elevated by $\Delta h = 30 \, \text{ft}$ . $E_p = (15 \, \text{slugs}) \cdot (32.174 \, \text{ft/s}^2) \cdot (30 \, \text{ft}) \approx 14{,}478 \, \text{ft·lb}$

Hydropower Potential Calculation

The potential energy stored in a reservoir or tank can be estimated for hydropower applications. The total energy is the integral of the energy contained in horizontal slices of water, which depends on the elevation of each slice above the outlet.

For a slice of water at elevation $z$ with volume $dV$ , the potential energy is $dE_p = \rho \cdot g \cdot z \cdot dV$ , where $\rho$ is water density. The total energy is: $E_p = \rho \cdot g \cdot \int_{0}^{H} z \cdot A(z) \, dz$ where $A(z)$ is the cross-sectional area at height $z$ , and $H$ is the total water height.

Engineering Approach: This integral is commonly solved numerically by dividing the reservoir into horizontal slices and summing the contributions. A spreadsheet calculator can automate this for various reservoir shapes (e.g., cylindrical, conical, irregular). You can adapt such a calculator by inputting the reservoir geometry and water level data.

Note on Hydraulic Head: The change in elevation $\Delta h$ is often referred to as the hydraulic head in hydropower contexts. The available power is then approximated as $P = \eta \cdot \rho \cdot g \cdot Q \cdot \Delta h$ , where $\eta$ is system efficiency and $Q$ is volumetric flow rate.