The second essential equation in physics: The intuition of a quantity that is conserved

The second equation of physics — In an isolated system, energy is not lost, it transforms. This photograph illustrates this fundamental principle of physics: the total mechanical energy (E_m), expressed in joules, is the sum of kinetic energy (E_k) and potential energy (E_p). The moving object — here a person on a swing — embodies this equation: at the highest point, potential energy dominates; at the lowest point, kinetic energy prevails. Movement thus becomes a dance between height and speed, between gravity and inertia.
Image source: astronoo.com (new window) — AI-generated image, public domain.

Scientific summary

The Astronoo article presents the second essential equation of physics, that of the conservation of mechanical energy in an isolated system. It traces the historical emergence of the concept of energy, from Jean Bernoulli to Max Planck, then explains the relation E_mec = E_k + E_p. The notions of kinetic energy and potential energy are defined, as well as their mutual conversions. The article justifies the use of the joule as the unit of measurement and clearly distinguishes energy from power, measured in watts. It finally emphasizes the universal character of this conservation principle. This summary highlights the theoretical foundations and practical applications of this fundamental law.

What is the second essential equation in physics and why is the conservation of mechanical energy so fundamental?

This Astronoo article is devoted to the second essential equation of physics, often presented as the law of conservation of mechanical energy. The direct question is: how can a quantity remain constant while movement, speed, and height constantly change? The answer lies in the relation E_mec = E_k + E_p = constant, valid in an isolated system. Kinetic energy, linked to speed and mass, and potential energy, linked to position or configuration, transform into one another without ever disappearing. A pendulum, a swing, or a free fall illustrate this energetic ballet. The article also traces the history of the concept, from the Bernoullis to James Prescott Joule and Max Planck. It explains why the joule is the appropriate unit for measuring energy, and why the watt measures power, that is, a rate of energy. This distinction is crucial for understanding energy balances. Finally, the article shows that this seemingly simple equation expresses one of the most universal principles of nature: energy is neither created nor destroyed, it transforms.

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Principle of conservation of mechanical energy

A quantity born from the history of science

The concept of energy, as we understand it today, has evolved over the centuries. It was long confused with force and power.
The word 'energy' appeared in 1717 in a letter written by Jean Bernoulli (1667-1748) to define mechanical work. However, it was not until the mid-19th century that the principle of conservation of energy was established. It was Max Planck (1858-1947) who understood the essential scope of this law in his 1887 book, "The Principle of Conservation of Energy."
The central idea is that energy is a constant of the universe, that it can neither be created nor destroyed, but that it takes different forms over time.

The fundamental equation of mechanics

The second essential equation in physics, often taught just after Newton's second law, is the equation of conservation of mechanical energy in an isolated system.

E_mec = E_k + E_p = constant

The components of mechanical energy

This relation distinguishes kinetic energy and potential energy.

• E_mec is the total mechanical energy of the system in joules.

• E_k = ½ mv² is the kinetic energy, where m is the mass of the object and v its speed. The contributions of Jean le Rond d'Alembert (1717-1783) and other scientists, such as Daniel Bernoulli (1700-1782), were crucial to the development of this formula.

• E_p = mgh is the potential energy, which depends on the nature of the force field considered (gravitational, elastic, etc.), where h is the height relative to a reference point and g is the acceleration due to gravity (9.806 65 m/s2). This formula is associated with the work of Isaac Newton (1643-1727).

What does E_mec = E_k + E_p tell us?

Conversion between potential and kinetic energy

Potential energy can be converted into kinetic energy and vice versa. For example, a falling object transforms its gravitational potential energy (mgh) into kinetic energy (½ mv²). Since energy is conserved, E_k + E_p = constant.
It is the joule that was chosen to measure the different types of energy.

A universal principle of conservation

This equation is crucial because it expresses the principle of conservation of energy, one of the most fundamental principles of physics. It shows that, in an isolated system (without exchange of energy with the outside), the total energy remains constant, even if it can transform from one form to another (for example, from potential energy to kinetic energy and vice versa).
James Prescott Joule (1818-1889) played a crucial role in establishing this principle, with his experiments on the conservation of thermal and mechanical energy.

Why use the Joule?

A unifying unit for energy

To measure a quantity of energy, the joule is the most appropriate unit. It gives us a "snapshot" of the energy present in a system. For example, the energy stored by a dam due to its position in the gravitational field (E_p = mgh) is a quantity of energy even though the water is at rest.
By combining information about mass, height, and acceleration due to gravity, one can calculate the gravitational potential energy stored. This energy represents the energetic potential of the water to do work (such as turning a turbine) when it descends under the effect of gravity.
The use of the Joule allows unifying the measurement of different forms of energy and simplifying calculations in physics.

Why not the Watt?

Power and energy: two distinct notions

The watt, which is a very common unit, particularly in electricity, does not measure the same thing as the joule.
It is the unit of power.
Power is the rate at which energy is transferred or transformed. In other words, the watt measures the quantity of energy per unit of time. For example, 1 kilowatt-hour.

The energy flow in question

Energy is a quantity, power is a rate.
For example, in a water pipe, the flow rate of water (in liters per second) corresponds to power, while the total quantity of water that comes out of the pipe (in liters) corresponds to energy.

1 watt corresponds to 1 joule per second. A 100-watt bulb consumes 100 joules of electrical energy every second.

References

FAQ: Everything you need to know about the conservation of mechanical energy

What is mechanical energy?

Mechanical energy is the sum of the kinetic energy and the potential energy of a system. It is expressed in joules and remains constant in an isolated system, even if its components vary.

Why is mechanical energy conserved?

Because the internal forces of the system transform potential energy into kinetic energy, and vice versa, without global loss. In an isolated system, no energy is exchanged with the outside.

What is the difference between kinetic energy and potential energy?

Kinetic energy depends on mass and speed (E_k = ½ mv²). Potential energy depends on position or configuration, for example E_p = mgh in a uniform gravitational field.

Why is energy measured in joules and not in watts?

The joule measures a quantity of energy, while the watt measures power, that is, a rate of energy per unit of time. A 100-watt bulb consumes 100 joules per second.

Is mechanical energy always conserved in reality?

No, it is conserved only in an isolated system without friction or external force. In the presence of friction, part of the mechanical energy is transformed into thermal energy, but the total energy remains constant.

What is the relation to Newton's second law?

The conservation of mechanical energy follows from Newton's laws when the forces are conservative. It offers an often simpler method for solving motion problems without calculating all the forces.