The conservation of energy states that energy can be neither created nor destroyed, only converted from one form to another: the total energy of an isolated system remains constant over time. It is a fundamental principle of physics, with no known exception.
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How to measure it in class
You never measure “energy” itself. You measure velocities and heights, calculate Ec and Ep from them, and check whether their sum is conserved. FizziQ’s video analysis module is the most direct tool.
Steps:
- Film a pendulum or a bouncing ball, with a graduated ruler in the frame to set the scale
- Track the position of the object’s center frame by frame in the Kinematics module
- Read the height h at each instant and compute Ep = mgh, taking the lowest point as the origin
- Compute the velocity between two neighboring points, then Ec = ½ m v²
- Enter Ec, Ep and Em = Ec + Ep in FizziQ’s spreadsheet and plot the three curves on the same graph
- Observe that Ec and Ep vary in opposition, while Em stays essentially constant - a fluctuation of a few percent comes from the tracking, not from the physics
Scientific activities on this topic
- Mechanical energy of a pendulum - follow the Ep ↔ Ec conversion during an oscillation
- Pendulum: the relation a = 2gh/r - recover with the accelerometer the speed at the lowest point predicted by the energy balance
- Coefficient of restitution of a ball - measure the fraction of mechanical energy lost at each bounce
- Elastic collision: Newton’s cradle - a case where kinetic energy is almost fully conserved
- Pole vault: energy balance - follow the chain kinetic energy → elastic energy of the pole → potential energy
- G-forces on a roller coaster - relate height lost and speed gained on a real track
- Gravitational assist - understand why a probe gains speed without an engine
- Moment of inertia of a cylinder - split the energy between translation and rotation
- Impact vibrations and seismics - trace where the energy of a crashing object goes
- Attenuation of a circular wave - watch the same energy spread over a growing surface
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A principle, not a theorem provable by experiment
The conservation of energy has never been “proved” by a single experiment: it is a principle, adopted because no observation has ever contradicted it. Each time a deficit seemed to appear, physicists preferred to postulate a new form of energy rather than abandon the principle - and they were right every time. The most famous example is β decay: energy was missing from the balance, and in 1930 Pauli proposed an invisible particle to carry it. The neutrino was detected twenty-six years later.
What is conserved, and for whom
The principle applies only to an isolated system, that is, one that exchanges neither work nor heat with its surroundings. No real laboratory system is perfectly isolated. In practice one writes a balance: ΔE(system) = energy received − energy given up. The most frequent mistake in class is not forgetting the principle, it is forgetting to say which system is being considered.
Total energy is not mechanical energy
This is the central confusion. Mechanical energy Em = Ec + Ep is conserved only if the only forces doing work are conservative (weight, elastic force, electrostatic force). As soon as friction, a soft impact or plastic deformation comes into play, Em decreases: ΔEm = W(non-conservative forces), a negative quantity. Total energy, however, does not change; it has merely changed form, into internal (thermal) energy. Saying “the energy was lost” is a misuse of language: it has been degraded.
Then why do we speak of an energy crisis?
If energy is conserved, how can we run short of it? Because not all forms are equal. The second law of thermodynamics says that conversion into heat is easy and one-way: heat cannot be fully converted back into work. What becomes scarce is not energy, but usable energy, the kind that is concentrated and ordered. This is a distinction students rarely grasp on their own, and it deserves to be stated explicitly.
A little history
The principle took shape in the mid-19th century, from several independent lines of work: Julius Robert von Mayer (1842), James Prescott Joule and his measurements of the mechanical equivalent of heat (1843-1850), and Hermann von Helmholtz, who gave its first general statement in 1847. Before that, heat and motion were treated as separate fields; the decisive contribution was to understand that they are the same quantity in two forms. In 1918, Emmy Noether established an even deeper result: the conservation of energy follows from the invariance of the laws of physics under translation in time.
Formula
Energy balance of a system, in its most general form:
E(total, initial) = E(total, final)
For a system exchanging with its surroundings (first law of thermodynamics):
ΔU = W + Q
Conservation of mechanical energy, only without non-conservative forces:
Em = Ec + Ep = constant, that is ½ m v² + m g h = constant
In the presence of friction:
ΔEm = W(non-conservative forces) < 0
where:
- E: total energy (J)
- ΔU: change in internal energy of the system (J)
- W: work exchanged (J)
- Q: heat exchanged (J)
- Ec: kinetic energy (J)
- Ep: potential energy (J)
- Em: mechanical energy (J)
- m: mass (kg)
- v: velocity (m·s⁻¹)
- g: gravitational field strength, about 9.81 m·s⁻² in France
- h: height relative to the chosen origin (m)
Application examples
- A 58 g tennis ball dropped from 1.00 m has Ep = 0.58 J. It rises back to about 0.55 m: 0.32 J of mechanical energy remains, that is 55%. The missing 45% heated the ball and the floor, and went off as sound.
- A hydroelectric dam converts the potential energy of water: 1 m³ of water (1,000 kg) falling 100 m releases mgh ≈ 0.98 MJ, or 0.27 kWh. The actual efficiency of a turbine exceeds 90%, one of the best of all machines.
- A 75 kg cyclist (bike included) climbing 200 m of elevation supplies at least 147 kJ, or 41 Wh - the equivalent of a cereal bar, which gives an idea of the modest efficiency of the human body (about 25%).
- Regenerative braking in an electric vehicle recovers part of the kinetic energy in electrical form instead of dissipating it as heat: about 60 to 70% in urban driving.
- A thermal power plant converts only about 40% of the fuel’s energy into electricity. The remaining 60% is not “lost” in the sense of conservation: it is released as heat, as the second law requires.
FAQ
Q: If energy is always conserved, why does my pendulum eventually stop? A: Because what is conserved is total energy, not mechanical energy. With each oscillation, part of the mechanical energy passes into the air and into the pivot as heat. A sufficiently sensitive thermometer would see the temperature rise by a tiny fraction of a degree. Nothing has disappeared, everything has changed form.
Q: How do we know the principle is true, since we never measure all the energy? A: We do not know it in the sense of a proof. It is a principle: it is accepted because no experiment has ever contradicted it in more than a century and a half, and because assuming it true has allowed predictions later verified, such as the neutrino. A principle is judged by its fruitfulness, not by a direct proof.
Q: What is the difference between conservation of energy and conservation of mechanical energy? A: The first is universal. The second is a special case valid only when no non-conservative force does work: no solid friction, no fluid friction, no soft impact. In practice, as soon as there is contact or air, mechanical energy decreases. This is the most common confusion in the first years of high school physics.
Q: A gravitational slingshot gives a probe speed without fuel. Where does the energy come from? A: From the planet. In the heliocentric reference frame, the probe gains exactly what the planet loses on its orbit. Since the planet is billions of billions of times more massive, its loss is undetectable. The balance is rigorously even; you simply have to remember to include the planet in the system.
Q: Can a machine be built that produces more energy than it consumes? A: No. This is called a perpetual motion machine of the first kind, and it is forbidden by the conservation of energy. All machines of this type proposed over three centuries have turned out to contain either a hidden energy source or a measurement error. Patent offices reject such applications without examining them.
Related concepts
Mechanical Energy - Kinetic Energy - Potential Energy - Elastic Energy - Friction Force - Elastic Collision - Inelastic Collision