Conservation of Mechanical Energy
Conservation of mechanical energy states that the mechanical energy of an isolated system remains constant without friction.
Learning Objective

Formulate the principle of the conservation of the mechanical energy
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.
Terms

frictional force
Frictional force is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other.

isolated system
A system that does not interact with its surroundings, that is, its total energy and mass stay constant.

conservation
A particular measurable property of an isolated physical system does not change as the system evolves.
Full Text
Conservation of mechanical energy states that the mechanical energy of an isolated system remains constant in time, as long as the system is free of all frictional forces. In any real situation, frictional forces and other nonconservative forces are always present, but in many cases their effects on the system are so small that the principle of conservation of mechanical energy can be used as a fair approximation. An example of a such a system is shown in . Though energy cannot be created nor destroyed in an isolated system, it can be internally converted to any other form of energy.
A Mechanical System
An example of a mechanical system: A satellite is orbiting the Earth only influenced by the conservative gravitational force and the mechanical energy is therefore conserved. This acceleration is represented by a green acceleration vector and the velocity is represented by a red velocity vector.
Derivation
Let us consider what form the workenergy theorem takes when only conservative forces are involved (leading us to the conservation of energy principle). The workenergy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy (KE). In equation form, this is:
If only conservative forces act, then W_{net}=W_{c}, where W_{c} is the total work done by all conservative forces. Thus, W_{c }= ΔKE. ,
Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy (PE). That is, W_{c} = −PE. Therefore,
This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,
where i and f denote initial and final values. This equation is a form of the workenergy theorem for conservative forces; it is known as the conservation of mechanical energy principle.
Remember that the law applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy (KE+PE). 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 Term Reference
 Law
 Appears in these related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 approximation
 Appears in these related concepts: Numerical Integration, The Discrete Fourier Transform, and Roundoff Error
 conservative force
 Appears in these related concepts: Gravity, Springs, and Fundamental Theorem for Line Integrals
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 force
 Appears in these related concepts: Force of Muscle Contraction, Force, and First Condition
 friction
 Appears in these related concepts: ProblemSolving With Friction and Inclines, Inelastic Collisions in Multiple Dimensions, and The First Law: Inertia
 gravitational force
 Appears in these related concepts: Newton and His Laws, Application of Bernoulli's Equation: Pressure and Speed, and Weight of the Earth
 kinetic
 Appears in these related concepts: Friction: Static, The Kinetic Molecular Theory of Matter, and Postmodernist Sculpture
 kinetic energy
 Appears in these related concepts: Solid Solubility and Temperature, Pressure, and Types of Energy
 potential
 Appears in these related concepts: What is Potential Energy?, Conservative and Nonconservative Forces, and Linear Expansion
 potential energy
 Appears in these related concepts: Problem Solving With the Conservation of Energy, Escape Speed, and Defining Graviational Potential Energy
 work
 Appears in these related concepts: Heat and Work, Free Energy and Work, and The First Law of Thermodynamics
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources: