Physics Formula: Conservation of Energy

The law of conservation of energy is one of the fundamental principles in physics. It states that energy cannot be created or destroyed—only transformed from one form to another or transferred from one object to another. This principle helps us understand the behavior of systems in classical mechanics, thermodynamics, and modern physics.

What is Energy?

Energy is the ability to do work. It exists in various forms such as kinetic energy, potential energy, thermal energy, chemical energy, and electrical energy. In any closed system, the total amount of energy remains constant, even though it may change from one form to another. Energy is a scalar quantity and can be measured in joules (J).

Forms of Mechanical Energy

Kinetic Energy (KE): Energy due to motion, given by the formula:
$$ KE = \frac{1}{2}mv^2 $$ where m is mass and v is velocity.
Potential Energy (PE): Energy stored due to position in a force field (e.g., gravity):
$$ PE = mgh $$ where m is mass, g is acceleration due to gravity, and h is height.

The Law of Conservation of Energy

According to this law, the total mechanical energy in a system (kinetic + potential) remains constant if only conservative forces (like gravity) are doing work. This principle is fundamental to our understanding of motion and energy transformations in mechanical systems.

$$ E_{\text{total}} = KE + PE = \text{constant} $$

This means that:

$$ \frac{1}{2}mv^2 + mgh = \text{constant} $$

If energy seems to disappear (for example, due to friction), it has simply changed form—like into heat—rather than being destroyed.

Conservative vs. Non-Conservative Forces

Understanding which forces are at play is key to applying the conservation of energy. Forces are categorized into two main types based on whether they conserve mechanical energy.

Conservative forces: Forces where the work done is independent of the path taken (e.g., gravity, spring force). These forces conserve mechanical energy.
Non-conservative forces: Forces where the work done depends on the path (e.g., friction, air resistance). These forces convert mechanical energy into other forms, such as heat or sound.

Modified Formula with Work

When non-conservative forces are involved, we account for the work they do on the system:

$$ E_{\text{initial}} + W_{\text{nc}} = E_{\text{final}} $$

Where $ W_{\text{nc}} $ is the work done by non-conservative forces. In this case, the energy is no longer entirely mechanical, as some of it is converted into other forms like heat or sound.

Example 1: Falling Object

Consider a ball of mass 2 kg dropped from a height of 10 meters. Find its velocity just before hitting the ground.

Solution:

Initial Potential Energy: $$ PE = mgh = 2 \times 9.8 \times 10 = 196 \, \text{J} $$
At the ground, PE = 0. All energy becomes KE: $$ KE = \frac{1}{2}mv^2 = 196 $$ Solving for $ v $: $$ v = \sqrt{\frac{2 \times 196}{2}} = \sqrt{196} = 14 \, \text{m/s} $$

Example 2: Pendulum Swing

A pendulum is raised to a height of 0.5 meters and released. What is its speed at the lowest point of its swing?

Solution:

At the top: PE = $ mgh $, KE = 0
At the bottom: PE = 0, all becomes KE
Using energy conservation: $$ mgh = \frac{1}{2}mv^2 $$ Cancel mass: $$ gh = \frac{1}{2}v^2 $$ $$ v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 0.5} = \sqrt{9.8} \approx 3.13 \, \text{m/s} $$

Application in Real Life

Roller Coasters

Roller coasters are an excellent real-world example of the law of conservation of energy. At the highest point of the coaster, the car has maximum potential energy due to its height. As it descends, this potential energy is converted into kinetic energy, increasing the speed of the car. At the lowest point, most of the potential energy has been converted into kinetic energy. Despite some energy being lost to friction, the total energy remains constant throughout the ride.

Hydroelectric Dams

Hydroelectric power plants take advantage of gravitational potential energy. Water stored in a reservoir has potential energy due to its height. When released, the water flows downward, gaining kinetic energy. This kinetic energy is used to turn turbines, which convert the energy into electrical power. In this process, the total mechanical energy of the water system is conserved, although some of it is lost as heat and other forms of energy.

Car Engines and Brakes

In a car engine, chemical energy from fuel is converted into kinetic energy that drives the car forward. When the car slows down or stops, the kinetic energy is converted into thermal energy due to friction in the brakes. While mechanical energy isn’t conserved in this case (since it’s converted to heat), the total energy in the system—including heat energy—is conserved.

Electric Generators

Electric generators work by converting mechanical energy into electrical energy. When a magnet spins inside a coil of wire, it induces an electric current, converting the mechanical energy of the rotating magnet into electrical energy. In this case, the law of conservation of energy still holds, as the total mechanical energy is converted into electrical energy.

Muscle Energy in Humans

Humans also rely on the conservation of energy for movement. Chemical energy stored in food is converted into mechanical energy in muscles. This mechanical energy is used for movement, while some of it is lost as heat. The human body operates similarly to an engine, with chemical energy being converted to kinetic energy, and some of it transformed into thermal energy.

Energy Diagrams

Energy diagrams are helpful tools for visualizing the conservation of energy. In simple systems, such as a mass-spring system or a falling object, an energy diagram can show how potential energy is converted into kinetic energy and vice versa. These diagrams can illustrate the point at which energy transformations occur, helping students and scientists understand how systems behave in a simplified manner.

Limitations and Considerations

While the law of conservation of energy is universally true, it’s important to consider certain limitations and factors in practical applications:

Friction and Heat: In real-world systems, friction is almost always present, which leads to the conversion of mechanical energy into heat. This makes it difficult to track energy in purely mechanical forms.
Entropy: According to the second law of thermodynamics, the total entropy of an isolated system will increase over time. While energy is conserved, it may become less useful over time due to entropy.
Quantum Systems: In quantum mechanics, energy conservation holds but in probabilistic terms. The behavior of particles can be uncertain, and energy fluctuations can occur at small scales.

Example 3: Spring System

A spring with constant $ k = 100 \, \text{N/m} $ is compressed by 0.2 m. How much energy is stored in the spring?

Solution:

Using the formula for elastic potential energy: $$ PE = \frac{1}{2}kx^2 = \frac{1}{2} \times 100 \times (0.2)^2 = 2 \, \text{J} $$

When released, this 2 J of energy becomes kinetic energy.

Example 4: Friction Involvement

A 1 kg object slides down a 5 m ramp inclined at 30°, with 2 J of energy lost to friction. What is the final speed at the bottom?

Solution:

Initial PE: $$ PE = mgh = 1 \times 9.8 \times 5 \times \sin(30^\circ) = 24.5 \, \text{J} $$
Work done by friction: 2 J
Energy available for KE: $$ KE = 24.5 - 2 = 22.5 \, \text{J} $$
Find velocity: $$ \frac{1}{2}mv^2 = 22.5 \Rightarrow v = \sqrt{2 \times 22.5} = \sqrt{45} \approx 6.7 \, \text{m/s} $$

Key Takeaways

Energy is never lost—only transformed.
Mechanical energy is conserved in systems with only conservative forces.
In presence of non-conservative forces, total energy still remains constant but is shared across different forms.
In complex systems, energy is distributed in forms such as heat, sound, and light.

Conclusion

The law of conservation of energy is essential in understanding physical processes, from simple mechanics to complex engineering systems. Its applications span across disciplines and technologies, reinforcing the idea that energy is the most persistent entity in the universe. Whether it’s the motion of a falling object, the workings of a roller coaster, or the function of a hydroelectric dam, the transformation and conservation of energy play a central role in shaping the world around us.

References

Halliday, Resnick, & Walker - Fundamentals of Physics
Serway & Jewett - Physics for Scientists and Engineers
OpenStax - Physics Textbook

Formula Quest Mania

Physics Formula: Conservation of Energy