Potential energy and conservation of energy

Stored energy

Potential energy is stored energy associated with position or arrangement. A book held above the floor has gravitational potential energy because gravity can pull it downward. A stretched rubber band has elastic potential energy because it can snap back. A compressed spring, water behind a dam, and a drawn bow all store energy that can later become motion.

Near Earth's surface, gravitational potential energy is often written:

$U_g = mgh$

Here $U_g$ is gravitational potential energy, $m$ is mass, $g$ is gravitational field strength, and $h$ is height above a chosen reference level. The higher the object is lifted, the more gravitational potential energy it has.

Choosing a reference level

Height depends on where you choose zero. A book may be 1 meter above the floor but 10 meters above the ground outside a window. Potential energy values depend on the chosen reference level, but changes in potential energy are what usually matter.

If a book falls from a table to the floor, its gravitational potential energy decreases. That lost potential energy becomes kinetic energy as the book speeds up, along with some sound and thermal energy when it hits.

Conservation of energy

The law of conservation of energy says energy cannot be created or destroyed in an isolated system. It can only be transferred or transformed from one form to another. This is one of the most important ideas in all of science.

For mechanical systems with little friction, we often use:

$E = K + U$

This means total mechanical energy is the sum of kinetic energy and potential energy. On a roller coaster, a car at the top of a hill has lots of gravitational potential energy and relatively little kinetic energy. As it rolls downward, potential energy decreases while kinetic energy increases. At the bottom, the car is moving fastest.

Energy with friction

Friction seems to make energy disappear because moving objects slow down. But energy is not destroyed. Friction transforms organized motion into thermal energy, sound, and microscopic motion of atoms in the surfaces.

For example, when you rub your hands together, friction converts motion into thermal energy, making your hands warmer. When brakes stop a bicycle wheel, kinetic energy becomes thermal energy in the brake pads, rims, tires, and air.

So conservation of energy still holds, but mechanical energy may not be conserved if energy is transformed into non-mechanical forms.

Elastic potential energy

Springs and stretchy materials can store elastic potential energy. A simple spring model is:

U_s = rac{1}{2}kx^2

Here $k$ is the spring constant and $x$ is how far the spring is stretched or compressed from its natural length. A stiffer spring has a larger $k$ and stores more energy for the same stretch.

Elastic energy appears in trampolines, bows, shock absorbers, mattresses, and many toys. As with gravitational energy, stored energy can become kinetic energy.

Why energy is powerful

Energy methods are useful because they often ignore unnecessary details. To estimate the speed of a roller coaster at the bottom of a hill, you may not need to know every force at every point. You can compare energy at the top and bottom.

Energy is also a unifying idea. Heat, light, motion, electricity, sound, chemical fuel, and nuclear reactions can all be discussed as energy transformations.

The big idea

Potential energy is stored energy. Kinetic energy is energy of motion. Conservation of energy says the total energy of an isolated system remains constant, even though energy changes form. This principle helps explain falling, bouncing, braking, heating, power generation, and much of everyday technology.