7.6 Conservation of Energy

Conservation of Energy

Energy conservation is one of the most powerful principles in all of physics. It states that energy cannot be created or destroyed, only converted from one form to another. If you can identify all the forms of energy in a system at one moment, you can predict what happens next, whether that's a ball falling, a spring launching a projectile, or a roller coaster cresting a hill.

This section covers the main forms of energy, how they transform into one another, how to track energy through a system using conservation equations, and how to account for real-world energy losses through efficiency.

Conservation of Energy Law

The total energy of an isolated system (one with no external forces doing work on it) remains constant over time. Energy can change forms, but the total amount never changes.

Written as an equation:

$KE_i + PE_i = KE_f + PE_f$

This says the sum of kinetic and potential energy at the start equals the sum at the end, as long as no energy leaves the system. If non-conservative forces like friction are present, you modify this to:

$KE_i + PE_i = KE_f + PE_f + W_{\text{friction}}$

where $W_{\text{friction}}$ is the energy lost to thermal energy through friction.

Example: A ball dropped from rest converts gravitational potential energy into kinetic energy as it falls. Right before hitting the ground, nearly all the potential energy has become kinetic energy. On impact, that kinetic energy converts into thermal energy and sound energy. At every stage, the total energy in the system is the same.

This principle lets you solve problems involving collisions, oscillations, projectiles, and more without needing to track forces at every instant.

Forms of Energy

Kinetic energy is the energy of motion:

$KE = \frac{1}{2}mv^2$

where $m$ is mass (kg) and $v$ is velocity (m/s). Double the speed and you quadruple the kinetic energy, which is why speed matters so much in car crashes.

Potential energy is stored energy due to an object's position or configuration:

• Gravitational potential energy: $PE_g = mgh$, where $m$ is mass, $g$ is the acceleration due to gravity ($9.8 \text{ m/s}^2$), and $h$ is the height above a chosen reference point. You get to pick where $h = 0$, so choose whatever makes the problem easiest.
• Elastic potential energy: $PE_e = \frac{1}{2}kx^2$, where $k$ is the spring constant (N/m) and $x$ is the displacement from the spring's equilibrium position.

Thermal energy is the energy associated with the random motion of particles in a substance. It's directly related to temperature. Friction converts kinetic energy into thermal energy, which is why a book sliding across a table slows down.

Electrical energy is associated with the movement of electric charges, either stored in electric fields or transferred through currents.

Chemical energy is stored in the bonds between atoms in molecules. It's released or absorbed during chemical reactions (for example, the chemical energy in gasoline converts to thermal and kinetic energy in a car engine).

Efficiency of Energy Conversions

No real-world energy conversion is perfect. Some energy always ends up as thermal energy due to friction, air resistance, or other irreversible processes. Efficiency measures how much of the input energy actually ends up doing what you want:

$\text{Efficiency} = \frac{\text{Useful work output}}{\text{Total energy input}} \times 100\%$

How to calculate efficiency:

1. Identify the total energy input to the system.
2. Identify the useful work output (the energy in the form you actually want).
3. Divide useful output by total input.
4. Multiply by 100% to get a percentage.

Example: An electric motor receives 100 J of electrical energy and performs 70 J of mechanical work. The remaining 30 J is lost as heat.

$\text{Efficiency} = \frac{70 \text{ J}}{100 \text{ J}} \times 100\% = 70\%$

A higher efficiency means less energy is wasted. In practice, you'll never reach 100% efficiency because some energy is always lost to thermal dissipation.

Work, Power, and Force in Energy Conservation

Work is the transfer of energy when a force causes a displacement:

$W = Fd\cos\theta$

where $F$ is the applied force, $d$ is the displacement, and $\theta$ is the angle between the force and the direction of displacement. When the force is perpendicular to the motion ($\theta = 90°$), no work is done. This is why a person carrying a box horizontally does no work on the box against gravity.

The work-energy theorem connects work directly to kinetic energy:

$W_{\text{net}} = \Delta KE = KE_f - KE_i$

The net work done on an object equals the change in its kinetic energy. This is one of the most useful problem-solving tools in this unit.

Power is the rate at which work is done:

$P = \frac{W}{t}$

where $W$ is work (in joules) and $t$ is time (in seconds). Power is measured in watts (W), where $1 \text{ W} = 1 \text{ J/s}$. Two machines can do the same total work, but the one that does it faster has greater power output.