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Conservation of energy review

Review the key concepts, equations, and skills for the conservation of energy, mechanical energy, and nonconservative work.

Key terms

Term (symbol)Meaning
Law of conservation of energyThe total energy of an isolated system is constant. Energy is neither created nor destroyed, it can only be transformed from one form to another or transferred from one system to another.
Mechanical energy (Em)Sum of the kinetic and potential energy. SI unit of joule (J).
Conservation of mechanical energy principleIf only conservative forces do work, the mechanical energy of a system is constant in any process.
Thermal energyInternal energy present in a system due to its temperature.
Nonconservative work (WNC)Work done by nonconservative forces. Example is work done by friction, which produces thermal energy. SI unit of joule (J).


EquationSymbol breakdownMeaning in words
Em=K+UEm is mechanical energy, K is kinetic energy, U is potential energy.The total mechanical energy of a system is the sum of the total kinetic energy and total potential energy.
K0+U0=K+UorΔK+ΔU=0K0 is initial kinetic energy, U0 is initial potential energy, K is final kinetic energy, U is final potential energy, ΔK is change in kinetic energy, and ΔU is change in potential energy.The initial mechanical energy of a system equals the final mechanical energy for a system where no work is done by non-conservative forces (conservation of mechanical energy principle).
K0+U0+WNC=K+UorWNC=ΔK+ΔUK0 is initial kinetic energy, U0 is initial potential energy, K is final kinetic energy, U is final potential energy, ΔK is change in kinetic energy, ΔU is change in potential energy, and WNC is work nonconservative.The change in mechanical energy of a system is equal to the total work done on the system by all nonconservative forces.

How to write the conservation of energy equation

The conservation of energy equation
is always true in any scenario. However, the conservation equation may look different depending on the problem because different forces and types of energy may be involved. To write the correct energy conservation equation:
  1. Draw a picture of the scenario, list your known information, and identify your system. Don’t forget that potential energy and work done by friction must include two objects.
  2. Decide what the initial and final locations will be for analyzing energy conservation by including our desired unknown in one of the locations and all the known information in the other location. Label the kinetic and potential energies at these two points.
  3. Designate the lower of the two positions as the zero height location. This eliminates the potential energy term for this location and simplifies our conservation of energy equation.
  4. If there are no nonconservative forces like friction, then use the conservation of mechanical energy:
Or if nonconservative forces are present, then include WNC with the final energies:
  1. Cancel out any of the energy terms that are zero to simplify your equation. For example, if the system has no motion at the final or initial positions, then remove the kinetic energy terms from the equation.

Common mistakes and misconceptions

  1. The conservation of energy equation only compares a system’s energy for the final and initial points in time. There may be different combinations of energy between these two points, but the equation we use only considers the final and initial energies.
For example, consider dropping a ball on a spring (see Figure 1 below). For the spring-mass-Earth system, we can analyze the energy from the moment of the ball’s drop (left side) to the point where the ball is at its lowest point on the spring (right side). It starts as all gravitational potential energy, transitions to a combination of kinetic and gravitational potential energy as the ball drops, and ends with only elastic potential energy.
The energy conservation equation for the ball-spring-earth system for its drop position and the maximum spring compression position is
Even though the ball is moving during the fall, the balls has no kinetic energy at the initial and final point.
Figure 1. Energy transformations of a ball dropped on a spring.
  1. People mistakenly think energy is constant for an object. The total energy of the universe is constant, but energy can be transferred between systems that we define in the universe. If one system gains energy, some other system must have lost energy to conserve the total energy in the universe.
An example of this would be pushing a friend on a sled. Your friend was initially at rest, but after the push he has kinetic energy. Your pushing force transferred energy to the friend.

Learn more

For deeper explanations of the law of conservation of energy, see our video about the law of conservation of energy and LOL diagrams.
To check your understanding and work toward mastering these concepts, check out our exercises on predicting changes in energy and using the conservation of energy to numerically solve for an unknown.

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