Gravitational potential energy is the stored energy objects possess due to their position in a gravitational field. It's crucial for understanding how energy transforms as objects move vertically, like in free fall or pendulum swings.

Calculating gravitational potential energy involves mass, height, and gravitational acceleration. This concept ties into the broader principle of energy conservation, helping us analyze various physical systems and predict object behavior in Earth's gravity.

Gravitational Potential Energy Fundamentals

Gravitational potential energy concept

• Gravitational potential energy (GPE) represents stored energy in objects due to their position within gravitational fields measured relative to a reference point (Earth's surface)
• Work done by gravity equals negative change in gravitational potential energy $W = -\Delta U_g$ demonstrating inverse relationship
• GPE decreases when gravity performs positive work (object falls) while GPE increases with negative work (object rises)

Derivation of potential energy expression

• Work-energy theorem states work done by a force equals change in energy
• Gravitational force near Earth's surface expressed as $F_g = mg$ where m is mass and g is gravitational acceleration (9.8 m/s²)
• Work done by gravity calculated as $W = F_g \cdot d = mgd$ where d is vertical displacement
• Change in gravitational potential energy derived as $\Delta U_g = -W = -mgd$
• Final expression for GPE near Earth's surface $U_g = mgh$ where h represents height above reference point

Applications and Problem Solving

Near-Earth potential energy calculations

• Key variables for GPE problems include mass (m), gravitational acceleration (g), and height (h)
• Common problem types involve calculating GPE at different heights, finding GPE changes for vertical displacements, and determining heights from given GPE values
• Ensure unit consistency using SI units (kg for mass, m for height, J for energy)

Mechanical energy conservation in gravity

• Mechanical energy combines kinetic and potential energies $E_{mech} = KE + PE$
• Conservation of energy principle states total mechanical energy remains constant without non-conservative forces
• Energy conversions occur between GPE and kinetic energy during motion (free fall, pendulum swings)
• Applications include analyzing pendulum motion, roller coaster dynamics, and projectile trajectories
• Non-conservative forces (friction, air resistance) cause gradual decrease in mechanical energy over time