Physics-X Study Notes on Potential Energy and Conservative Forces

Examples of Potential Energy Due to Position

• Body on a Building Roof: A body lying on the roof of a building possesses potential energy. When this body is allowed to fall down, it has the capacity to perform work.
• Water Stored in Dams: Water stored at great heights in dams possesses potential energy. This energy is utilized to run turbines for the purpose of generating hydroelectricity.

Examples of Potential Energy Due to Configuration

• Wound Spring in a Toy Car: In a toy car, a wound spring has potential energy. When the spring is released, this potential energy transforms into kinetic energy, which causes the toy car to move.
• Stretched Bow: A stretched bow possesses potential energy. Upon release, it shoots an arrow in the forward direction with high velocity. During this process, the potential energy of the stretched bow is converted into kinetic energy.
• Compressed Spring in a Loaded Gun: Due to the potential energy stored in the compressed spring of a loaded gun, a bullet is fired with a large velocity when the gun is triggered.

Classification of Potential Energy Types

• Gravitational Potential Energy: This is the potential energy associated with the state of separation of two bodies that attract one another through gravitational force.
• Elastic Potential Energy: This refers to the potential energy associated with the state of compression or extension of an elastic object, such as a spring.
• Electrostatic Potential Energy: This is the energy resulting from the interaction between two electric charges.

Gravitational Potential Energy (GPE)

• Definition: Gravitational potential energy is the energy possessed by a body by virtue of its position above the surface of the earth.
• Reference Point: At the surface of the earth, the height is considered zero ($h = 0$). Consequently, the gravitational potential energy at the earth's surface is zero ($U = 0$).

Mathematical Expression and Derivation for GPE

• Considerations and Assumptions:
  • Let a body have a mass of $m$.
  • The body is situated at a height $h$ above the earth's surface.
  • Let $g$ represent the acceleration due to gravity at that specific location.
  • For heights much smaller than the radius of the earth ($h \ll R$), the value of $g$ can be taken as a constant.
• Derivation Steps:
  • The force needed to lift the body up with zero acceleration is equal to the weight of the body:         $F = \text{Weight of the body} = mg$
  • The work done on the body in raising it through height $h$ is calculated as:         $W = F \times h = mg \times h$
  • This work done against gravity is stored as the gravitational potential energy ($U$) of the body:         $U = mgh$

Conservative and Non-Conservative Forces

• Definition of Conservative Force: A force is defined as conservative if the work done by the force in displacing a particle from one point to another is independent of the path followed by the particle and depends only on the endpoints.
• Path Independence: Consider a particle moving from point $A$ to point $B$ along either path 1 or path 2 (as illustrated in Fig. 6.16a). If a conservative force $F$ acts on the particle, the work done on the particle is identical along both paths.
  • Mathematically expressed as:         $W_{AB} \text{ (along path 1)} = W_{AB} \text{ (along path 2)}$
• Round Trip Properties: Suppose the particle moves in a round trip from point $A$ to point $B$ along path 1 and then returns to point $A$ along path 2 (as illustrated in Fig. 6.16b).
  • For a conservative force, the work done on the particle along the path from $A$ to $B$ is equal to the negative of the work done on the particle along path 2 from $B$ to $A$.
  • Mathematically:         $W_{AB} \text{ (along path 1)} = -W_{BA} \text{ (along path 2)}$