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## High school physics

### Course: High school physics>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, start subscript, start text, m, end text, end subscript)Sum of the kinetic and potential energy. SI unit of joule (start text, J, end text).
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, start subscript, start text, N, C, end text, end subscript)Work done by nonconservative forces. Example is work done by friction, which produces thermal energy. SI unit of joule (start text, J, end text).

## Equations

EquationSymbol breakdownMeaning in words
E, start subscript, start text, m, end text, end subscript, equals, K, plus, UE, start subscript, start text, m, end text, end subscript 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{aligned} K_0+U_0 &= K+U \\ &\text {or} \\ \Delta K + \Delta U &= 0\end{aligned}K, start subscript, 0, end subscript is initial kinetic energy, U, start subscript, 0, end subscript is initial potential energy, K is final kinetic energy, U is final potential energy, delta, K is change in kinetic energy, and 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{aligned} K_0+U_0+W_{\text{NC}} &= K+U \\ &\text {or} \\ W_{\text {NC}} &= \Delta K + \Delta U\end{aligned}K, start subscript, 0, end subscript is initial kinetic energy, U, start subscript, 0, end subscript is initial potential energy, K is final kinetic energy, U is final potential energy, delta, K is change in kinetic energy, delta, U is change in potential energy, and W, start subscript, start text, N, C, end text, end subscript 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, start subscript, 0, end subscript, plus, U, start subscript, 0, end subscript, plus, W, start subscript, start text, N, C, end text, end subscript, equals, K, plus, 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, start subscript, 0, end subscript, plus, U, start subscript, 0, end subscript, equals, K, plus, U
Or if nonconservative forces are present, then include W, start subscript, start text, N, C, end text, end subscript with the final energies:
K, start subscript, 0, end subscript, plus, U, start subscript, 0, end subscript, equals, K, plus, U, plus, W, start subscript, start text, N, C, end text, end subscript
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, start subscript, start text, g, comma, 0, end text, end subscript, equals, U, start subscript, start text, s, end text, end subscript
Even though the ball is moving during the fall, the balls has no kinetic energy at the initial and final point.
On the left is a ball above a spring. Next to the ball is a label of v_o=0. On the right side the ball is on the spring and compressing it. Next to the ball is a label of v=0. In the middle is a series of green energy labels. At the top is U_g, next down is U_g and K, next down is U_g, U_s, and K, at the bottom is U_s.
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.