5.12 Potential Energy (P.E.)

Potential energy is the stored energy an object possesses due to its position, shape, or state. It is the "potential" for an object to do work. There are several forms of potential energy, with gravitational and elastic being the most common in mechanics.

Gravitational Potential Energy

This is the energy an object has due to its position in a gravitational field, specifically its height above a reference point.

• Definition: The stored energy resulting from an object's vertical position or height.

• Formula: $P.E.=mgh$ Where:

  • $m$ is the mass of the object (kg)
  • $g$ is the acceleration due to gravity ($\approx 9.8{\text{ ms}}^{-2}$ on Earth)
  • $h$ is the height of the object relative to a chosen zero reference level

• Key Characteristics:

  • The choice of the zero level (e.g., the floor, the ground) is arbitrary. P.E. is a relative quantity; only the change in potential energy ($\Delta P.E.=mg\Delta h$) is physically significant.
  • Lifting an object against gravity increases its gravitational potential energy.

Example: A book held 2 m above the ground with mass 1 kg has $P.E.=1\times 9.8\times 2=19.6\text{ J}$. If released, this potential energy converts into kinetic energy as it falls.

Elastic Potential Energy

This is the energy stored in an elastic object, like a spring or a rubber band, when it is stretched or compressed.

• Definition: The stored energy resulting from the deformation of an elastic object.
• The amount of stored energy depends on the object's stiffness (spring constant $k$) and the distance $x$ it is stretched or compressed from its equilibrium (natural) position: $P.E{.}_{elastic}=\frac{1}{2}k{x}^2$

Example: A spring with $k=200{\text{ Nm}}^{-1}$ compressed by $x=0.1\text{ m}$ stores $P.E.=\frac{1}{2}(200)(0.1{)}^2=1\text{ J}$.

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Conservative and Non-Conservative Forces

This distinction is fundamental to understanding when potential energy can be defined.

Conservative Forces

A conservative force is one for which:

1. The work done is independent of the path taken between two points.
2. The work done over any closed path is zero.
3. The energy stored can be fully recovered — it can be converted back to kinetic energy without loss.

Examples of conservative forces:

• Gravitational force
• Elastic (spring) force
• Electrostatic force

Because these forces are conservative, we can define a potential energy associated with them. The work done by a conservative force equals the decrease in potential energy: $W_{conservative}=-\Delta P.E.$

Non-Conservative Forces

A non-conservative force is one for which:

1. The work done depends on the path taken.
2. Energy is dissipated (converted to heat, sound, etc.) and cannot be fully recovered.
3. No potential energy can be defined for these forces.

Examples of non-conservative forces:

• Friction
• Air resistance (drag)
• Tension in an inelastic string

Example: Sliding a book across a rough table from A to B via a straight path vs. a curved path — friction does different amounts of work along each path, and the energy is lost as heat.

Summary Table

Why This Matters

Only conservative forces allow us to define potential energy. When only conservative forces act on a system, mechanical energy (K.E. + P.E.) is conserved. When non-conservative forces (like friction) act, mechanical energy decreases — the "lost" energy appears as heat or sound.