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### Course: AP®︎/College Physics 1>Unit 5

Lesson 6: Conservation of energy

# 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 (${E}_{\text{m}}$)Sum of the kinetic and potential energy. SI unit of joule ($\text{J}$).
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 (${W}_{\text{NC}}$)Work done by nonconservative forces. Example is work done by friction, which produces thermal energy. SI unit of joule ($\text{J}$).

## Equations

EquationSymbol breakdownMeaning in words
${E}_{\text{m}}=K+U$${E}_{\text{m}}$ is mechanical energy, $K$ is kinetic energy, $U$ is potential energy.The total mechanical energy of a system is the sum of the total kinetic energy and total potential energy.
$\begin{array}{rl}{K}_{0}+{U}_{0}& =K+U\\ & \text{or}\\ \mathrm{\Delta }K+\mathrm{\Delta }U& =0\end{array}$${K}_{0}$ is initial kinetic energy, ${U}_{0}$ is initial potential energy, $K$ is final kinetic energy, $U$ is final potential energy, $\mathrm{\Delta }K$ is change in kinetic energy, and $\mathrm{\Delta }U$ 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).
$\begin{array}{rl}{K}_{0}+{U}_{0}+{W}_{\text{NC}}& =K+U\\ & \text{or}\\ {W}_{\text{NC}}& =\mathrm{\Delta }K+\mathrm{\Delta }U\end{array}$${K}_{0}$ is initial kinetic energy, ${U}_{0}$ is initial potential energy, $K$ is final kinetic energy, $U$ is final potential energy, $\mathrm{\Delta }K$ is change in kinetic energy, $\mathrm{\Delta }U$ is change in potential energy, and ${W}_{\text{NC}}$ 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
${K}_{0}+{U}_{0}+{W}_{\text{NC}}=K+U$
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:
${K}_{0}+{U}_{0}=K+U$
Or if nonconservative forces are present, then include ${W}_{\text{NC}}$ with the final energies:
${K}_{0}+{U}_{0}=K+U+{W}_{\text{NC}}$
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
${U}_{\text{g,0}}={U}_{\text{s}}$
Even though the ball is moving during the fall, the balls has no kinetic energy at the initial and final point.
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.

## Want to join the conversation?

• can there be a video showing examples for the numerical calculations?
(29 votes)
• can someone upload a video explains this a little more with detail and examples?
(6 votes)
• The intial amount of energy in a closed system is equal to the final amount of energy. Energy can't be created nor destroyed, its always transferred.
(2 votes)
• Can someone explain the difference between the formulas
K0 + U0 + Wnc = K + U
and K0 + U0 = K + U?
(1 vote)
• The formula K0 + U0 = K + U is a simplification of the formula K0 + U0 + Wnc = K + U. The first formula is used when there are only conservative forces present (such as elastic and gravitational potential) whereas the other formula is used when there are also non-conservative forces present such as friction. In this formula: (K0 + U0 = K + U) Wnc is equal to 0 and is therefore omitted from the equation
(2 votes)
• are non-conservative forces on the side with initial mechanical energy or the side with final mechanical energy?
(1 vote)
• Usually, non-conservative forces like friction or air resistance are on the side of the final energy, as they normally are taking away some energy from the system.
(2 votes)
• An object is dropped from rest. The law of conservation of energy provides a method for finding an approximate value for the speed with which the object hits the ground.

How does the law of conservation of energy help to approcimate the value for the speed of the object?
(1 vote)
• you can simply the law of conservation of energy to get the velocity
(1 vote)
• can there be a video showing you examples of numerical claculaions?
(1 vote)
• If there were no non-conservative forces, which there always are but if there were none, would a ball from a certain height just bounce on a hard surface to its first height? Because the energy is conserved within the system?
(1 vote)
• why must potential energy include two objects?
I mean, the gravitational potential energy of a ball can increase as its height increases without the need for another object
(1 vote)
• If there were no non-conservative forces, which there always are but if there were none, would a ball from a certain height just bounce on a hard surface to its first height? Because the energy is conserved within the system?
(1 vote)