Potential Energy - Physics

1. Fundamental Concepts

Definition: Potential energy is the stored energy an object has due to its position or configuration.
Gravitational Potential Energy: The potential energy due to an object's vertical position in a gravitational field, given by $$U = m \cdot g \cdot h$$ where $m$ is mass, $g$ is the acceleration due to gravity, and $h$ is height.
Elastic Potential Energy: The potential energy stored in a deformed elastic object, given by $$U = \frac{1}{2} k \cdot x^2$$ where $k$ is the spring constant and $x$ is the displacement from equilibrium.

2. Key Concepts

Conservation of Energy: In a closed system, the total energy remains constant; potential energy can be converted to kinetic energy and vice versa.

Reference Point: Potential energy is relative to a chosen reference point (often ground level).

Application: Used in physics to analyze systems involving changes in position or deformation.

3. Examples

Example 1 (Basic)

Problem: Calculate the gravitational potential energy of a 5 kg object at a height of 10 meters.

Step-by-Step Solution:

Use the formula for gravitational potential energy: $$U = m \cdot g \cdot h$$
Substitute the values: $$U = 5 \cdot 9.8 \cdot 10$$
Calculate: $$U = 490 \text{ Joules}$$

Validation: Substitute $m=5$, $g=9.8$, $h=10$ → Original: $5 \cdot 9.8 \cdot 10 = 490$; Simplified: $490$ ✓

Example 2 (Intermediate)

Problem: A spring with a spring constant of 200 N/m is compressed by 0.5 meters. Find the elastic potential energy stored in the spring.

Step-by-Step Solution:

Use the formula for elastic potential energy: $$U = \frac{1}{2} k \cdot x^2$$
Substitute the values: $$U = \frac{1}{2} \cdot 200 \cdot 0.5^2$$
Calculate: $$U = 25 \text{ Joules}$$

Validation: Substitute $k=200$, $x=0.5$ → Original: $\frac{1}{2} \cdot 200 \cdot 0.5^2 = 25$; Simplified: $25$ ✓

4. Problem-Solving Techniques

Visual Strategy: Draw diagrams to represent the system and identify all forces acting on the object.
Error-Proofing: Always check units and ensure they are consistent throughout the calculation.
Concept Reinforcement: Relate potential energy concepts to real-world examples, such as a roller coaster or a stretched rubber band.