Module 2 Potential Energy and Conservation of Energy

 

What is Potential Energy?

Potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position.

Learning Objectives

Relate the potential energy and the work

Key Takeaways

Key Points

  • If the work for an applied force is independent of the path, then the work done by the force is evaluated at the start and end of the trajectory of the point of application. This means that there is a function U(x), called a ” potential “.
  • It is tradition to define the potential function with a negative sign so that positive work is represented as a reduction in the potential.
  • Every conservative force gives rise to potential energy. Examples are elastic potential energy, gravitational potential energy, and electric potential energy.

Key Terms

  • Coulomb force: the electrostatic force between two charges, as described by Coulomb’s law
  • potential: A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.

Potential energy is often associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass of an object is performed by an external force that works against the force field of the potential. This work is stored in the force field as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to its initial position, reducing the stretch of the spring or causing the body to fall. The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position.

 

Examples of Potential Energy

There are various types of potential energy, each associated with a particular type of force. More specifically, every conservative force gives rise to potential energy. For example, the work of an elastic force is called elastic potential energy; work done by the gravitational force is called gravitational potential energy; and work done by the Coulomb force is called electric potential energy.

Gravity

Gravitational energy is the potential energy associated with gravitational force, as work is required to move objects against gravity.

Learning Objectives

Generate an equation that can be used to express the gravitational potential energy near the earth

Key Takeaways

Key Points

  • Gravitational potential energy near the earth can be expressed with respect to the height from the surface of the Earth as PE = mgh. g = gravitational acceleration (9.8m/s2). Near the surface of the Earth, g can be considered constant.
  • Over large variations in distance, the approximation that g is constant is no longer valid and a general formula should be used for the potential. It is given as: [latex]\text{U}(\text{r}) = \int_{\text{r}} (\text{G} \frac{\text{m} \text{M}}{\text{r}'^2}) \text{dr}' = -\text{G} \frac{\text{m} \text{M}}{\text{r}}\ + \text{K}.[/latex].
  • Choosing the convention that the constant of integration K=0 assumes that the potential at infinity is defined to be 0.

Key Terms

  • conservative force: A force with the property that the work done in moving a particle between two points is independent of the path taken.

Gravitational energy is the potential energy associated with gravitational force (a conservative force), as work is required to elevate objects against Earth’s gravity. The potential energy due to elevated positions is called gravitational potential energy, evidenced, for example, by water held in an elevated reservoir or behind a dam (as an example, shows Hoover Dam). If an object falls from one point to another point inside a gravitational field, the force of gravity will do positive work on the object, and the gravitational potential energy will decrease by the same amount.

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Hoover Dam: Hoover dam uses the stored gravitational potential energy to generate electricity.

Potential Near Earth

Gravitational potential energy near the Earth can be expressed with respect to the height from the surface of the Earth. (The surface will be the zero point of the potential energy. ) We can express the potential energy (gravitational potential energy) as:

[latex]\text{PE} = \text{m} \text{g} \text{h}[/latex],

where PE = potential energy measured in joules (J), m = mass of the object (measured in kg), and h = perpendicular height from the reference point (measured in m); g = gravitational acceleration (9.8m/s2). Near the surface of the Earth, g can be considered constant.

 

Conservation of Mechanical Energy

Conservation of mechanical energy states that the mechanical energy of an isolated system remains constant without friction.

Learning Objectives

Formulate the principle of the conservation of the mechanical energy

Key Takeaways

Key Points

  • The conservation of mechanical energy can be written as “KE + PE = const”.
  • Though energy cannot be created nor destroyed in an isolated system, it can be internally converted to any other form of energy.
  • In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between KE and various types of PE, with the total energy remaining constant.

Key Terms

  • conservation: A particular measurable property of an isolated physical system does not change as the system evolves.
  • isolated system: A system that does not interact with its surroundings, that is, its total energy and mass stay constant.
  • frictional force: Frictional force is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other.

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