8.1 Potential Energy of a System

Potential energy is stored energy that depends on an object's position or configuration. Work and potential energy are closely linked: work done on an object can change the potential energy of the system.

Different types of potential energy include gravitational and elastic potential energy. Their formulas help track energy transformations in physical systems, and the choice of reference point keeps potential energy calculations consistent.

Potential Energy and Work

Work and potential energy changes

• In a frictionless system, the work done on a particle equals the change in the particle's potential energy, known as the work-energy theorem
• Mathematically expressed as $W = \Delta U$, where $W$ represents the work done and $\Delta U$ represents the change in potential energy
• Positive work done on a particle increases its potential energy (lifting an object against gravity)
• Negative work done on a particle decreases its potential energy (lowering an object with gravity)
• In the absence of non-conservative forces (friction), the total mechanical energy (kinetic energy + potential energy) remains constant, following the principle of conservation of mechanical energy

Reference points for potential energy

• Potential energy is a relative quantity that depends on the choice of a reference point or zero level where the potential energy is defined to be zero
• The choice of reference point does not affect the behavior of the system or the calculations involving changes in potential energy since changes in potential energy ($\Delta U$) are independent of the reference point
• Common choices for reference points include:
  • Ground level or the lowest point in the system for gravitational potential energy
  • The unstretched or equilibrium position of the spring for elastic potential energy
• Maintaining consistency in choosing a reference point keeps system analysis and potential energy comparisons accurate

Types of Potential Energy

Formulas for gravitational and elastic energy

• Gravitational potential energy near Earth's surface:
  • Formula: $U_g = mgh$, where $m$ represents the mass of the object, $g$ represents the acceleration due to gravity (approximately $9.8 \text{ m/s}^2$ near Earth's surface), and $h$ represents the height of the object above the reference point
  • As an object moves higher above the reference point (typically ground level or the lowest point in the system), its gravitational potential energy increases
• Elastic potential energy in springs:
  • Formula: $U_s = \frac{1}{2}kx^2$, where $k$ represents the spring constant (a measure of the spring's stiffness) and $x$ represents the displacement of the spring from its equilibrium position
  • As a spring is compressed or stretched from its equilibrium position (the reference point where the spring is neither compressed nor stretched), its elastic potential energy increases
  • The spring constant $k$ determines how much force is required to compress or stretch the spring by a given distance (a higher $k$ value indicates a stiffer spring)

Forces and Energy in Systems

Conservative and non-conservative forces

• Conservative forces (e.g., gravitational force) are associated with potential energy and do work that is independent of the path taken
• Non-conservative forces (e.g., friction) dissipate energy and their work depends on the path taken
• The total energy of a system remains constant when only conservative forces are present

Potential energy and equilibrium

• Potential energy is related to the force acting on an object through its gradient
• The equilibrium position of a system corresponds to a minimum in the potential energy
• Displacement from equilibrium results in a restoring force that tends to return the system to its equilibrium state

Potential wells

• A potential well is a region where the potential energy is lower than the surrounding areas
• Objects tend to move towards and become trapped in potential wells, which can be visualized as valleys in potential energy diagrams