What is energy, and can it disappear?

Pull a ball on a string aside and let go. At first it swings briskly, then rises less and less, and finally comes to rest. The motion has gone. What about the energy? If energy is conserved, why does the pendulum not swing forever?

When friction and air resistance bring a pendulum to rest, its mechanical energy decreases, but energy is not destroyed: it passes to other parts of the system and takes other forms. To see conservation, we must also count what is no longer visible in the bob’s motion.

Energy is a physical quantity we use to describe the state of objects and their interactions numerically, and to track transformations. It is not a separate substance inside things. There are different ways to calculate it for motion, interaction with Earth and the microscopic state of matter; the results share a common unit, the joule, J. Richard Feynman’s lecture on energy explains this numerical character of the concept.

First, draw a boundary around the system

Imagine a boundary around whatever we are studying. Inside is the system; outside are its surroundings. The boundary can be imaginary: accounting for energy does not require building a box.

If we track only the bob, we leave out Earth, the air, the string and the support. When a hand lifts the bob aside, it transfers energy into the chosen system. When the bob pushes the air, energy can leave it. Transferring energy by a force acting through a displacement is called work. An individual object’s energy therefore does not have to stay constant.

For our pendulum, a wider system is useful: the bob, Earth, string, support and surrounding air. Much of what was an exchange with the surroundings now becomes redistribution within the system. We start counting after releasing the bob: the hand’s earlier work is already represented by the initial energy.

The balance rule is simple: the change in a system’s energy equals energy entering minus energy leaving. If the system is isolated, meaning that it exchanges no energy with its surroundings, its total energy stays constant. In a real room, sound, radiation or energy carried by moving air can cross the boundary; we must include them if they matter. OpenStax: energy exchange and the first law of thermodynamics.

Height is exchanged for speed

First, remove friction at the pivot and air resistance in our imagination. The string is light and inextensible, and its upper end is fixed. This ideal pendulum helps us separate one process from another.

The raised bob and Earth have more gravitational potential energy than when the bob is at the bottom. This is energy of their interaction, depending on their relative positions. We choose the lowest point of the path as the zero of potential energy.

The bob falls along its arc, and gravity speeds it up. Potential energy decreases while kinetic energy, energy of motion, increases. The speed is greatest at the bottom. The bob then rises on the other side: kinetic energy decreases and potential energy increases. Under our ideal conditions, it returns to the original height. OpenStax: potential energy of a system.

The sum of kinetic and potential energy is called mechanical energy. In the ideal pendulum this sum stays constant. At the highest point, the bob is momentarily still, but its mechanical energy is not zero: the potential part remains. Even an absence of motion does not mean an absence of energy.

Friction takes energy out of the visible swinging

Now restore the air and the real pivot. The bob sets air in motion, while matter at the rubbing surfaces deforms and interacts. The pendulum’s coordinated motion gradually transfers energy into many small movements and interactions. Some can first become energy of sound or air currents, which later dissipates too.

Internal energy, the energy of microscopic motions and interactions among particles of matter, increases. The change is spread among the bob, support and air; it may be too small for a hand to detect any warming. Less energy remains in the mechanical swinging, so the next high point is lower. Conservation of total energy does not require this mechanical share to be conserved.

The words matter here. Heat is energy transferred because of a temperature difference; internal energy is a stored quantity describing the system’s state. Friction can increase internal energy through a mechanical process without an initial temperature difference. A warmer support can subsequently transfer energy to cooler surroundings by heat transfer. The everyday phrase “energy turned into heat” hides these two different steps.

This distinction has an experimental basis. In Joule’s experiments, descending weights turned paddles in water, and a thermometer recorded its temperature increasing. A mechanical process changed the water’s internal energy; it was not being warmed by a hotter object. OpenStax: internal energy, heat and Joule’s experiments.

Compare two bottom crossings below: the ideal case and an illustrative case with mechanical energy losses. Does the full energy bar get shorter when its motion segment shrinks? The values 0.70 and 0.30 J were chosen to make the balance clear; they are neither measurements of a real pendulum nor a calculation of the drag force.

One initial joule: where it goes

Each full bar represents 1 J. Position sketches are not to scale.

Start: released from a height

Potential: 1.00 J
Kinetic: 0.00 J
Internal energy increase: 0.00 J
Total: 1.00 J

The speed is still zero.

Ideal pendulum: at the bottom

Potential: 0.00 J
Kinetic: 1.00 J
Internal energy increase: 0.00 J
Total: 1.00 J

The initial store is now energy of motion.

With resistance: first bottom crossing

Potential: 0.00 J
Kinetic: 0.70 J
Internal energy increase: 0.30 J
Total: 1.00 J

Chosen condition: 0.30 J has already dissipated.

With resistance: after stopping

Potential: 0.00 J
Kinetic: 0.00 J
Internal energy increase: 1.00 J
Total: 1.00 J

The initial store is now an increase in internal energy.

System: bob, Earth, string, support and surrounding air. No energy crosses the outer boundary in this diagram; sound and air flows have already dissipated. Potential energy is zero at the bottom. We show the extra energy from this release, not all the energy of the matter.

0.70/0.30 is an illustrative balance, not a measurement or a simulation of drag. The last card belongs to the case with resistance; the ideal pendulum does not stop.

In the last state the bob is at rest at the bottom, so both mechanical contributions are zero relative to our chosen reference. The extra energy that started the swinging remains in the system as an increase in internal energy. We are not claiming that the room’s initial internal energy was zero: the bars show only changes associated with this release.

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What we can calculate for one release

Take a bob of mass $m=0.20$ kg, raised through a vertical height $h=0.50$ m above the lowest point. Near Earth’s surface, round gravitational acceleration to $g\approx10$ m/s². Its initial energy relative to the bottom is then:

\begin{aligned} E_0&=mgh\\ &\approx0.20\cdot10\cdot0.50\\ &=1\ \text{J}. \end{aligned}

This approximation works for heights small compared with Earth’s radius; it does not describe lifting something across interplanetary distances. Feynman: limits of the gravitational height approximation.

Kinetic energy is $K=mv^2/2$ , where $v$ is speed relative to the room. Without friction, $K=E_0$ at the bottom, giving $v=\sqrt{2gh}\approx3.2$ m/s. In the illustrative second case, internal energy has already increased by 0.30 J before the bottom crossing. Then $K=1-0.30=0.70$ J and $v=\sqrt{2K/m}\approx2.6$ m/s. OpenStax: kinetic energy and its units.

Both energy formulas produce kg·m²/s², or joules. We found speed from a specified balance; we did not determine how long the motion takes or how resistance depends on speed. With zero energy transferred into internal energy, the ideal result returns; with all the initial energy transferred, none remains for motion. OpenStax: the energy approach to a pendulum.

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Why conserved energy is not always easy to use

Release a raised bob and you immediately obtain clear motion. The same additional energy, spread among the room’s enormous number of particles, does not restart that motion by itself. The same number of joules does not mean the same opportunity to obtain useful work.

Conservation gives us a balance, but by itself does not determine the direction of every process. The second law of thermodynamics adds a restriction: no engine can operate in a cycle, take heat only from a single source and turn it entirely into work without any other change. This does not forbid internal energy from becoming mechanical energy in general; it explains why “all the joules are still there” does not promise a lossless return to useful motion. OpenStax: statements of the second law.

Light can carry energy across the boundary too

Transferring energy does not necessarily require touching an object or bringing matter to it. Solar radiation crosses almost empty space, reaches Earth, and can increase the surface’s internal energy when absorbed. Visible light and infrared radiation are forms of electromagnetic radiation, which needs no material medium to propagate. OpenStax: energy transfer by radiation.

If our system is just the illuminated object, light brings energy in from outside. If the object radiates energy outwards, its stored energy can decrease. We neglected this exchange in the pendulum diagram; for a planet warming or a spacecraft cooling, it can be central.

Check your understanding: after a pendulum is released, sound carries some energy beyond the boundary of the chosen room. Must the sum of the energies inside the room stay the same?

No. The room in this description exchanges energy with its surroundings. To complete the balance, we must add the energy carried away by the sound wave, or widen the system to include the medium that received it. Saying “conserved” does not let us skip a flow through the boundary.

The pendulum stopped because its initial energy was no longer energy of its visible swinging. To check whether anything really “disappeared”, name the system, count other energy forms, and account for energy entering and leaving. The Solar System offers examples both of moving bodies and of energy carried by sunlight. Later in the course, we will explore light itself and the speed at which it travels.