Conservation of Mechanical Energy

Mechanical Energy and Its Two Parts

Mechanical energy is the total of an object's kinetic energy and its potential energy. Kinetic energy is the energy an object has because it is moving, and potential energy is the energy stored in a configuration, such as height in a gravitational field or stretch in a spring. We write

$E = K + U$

where $E$ is the total mechanical energy, $K$ is the kinetic energy and $U$ is the potential energy. Both parts are defined and developed in the articles on kinetic and potential energy, so here we only need their roles: $K$ depends on speed, and $U$ depends on position.

The key observation is that these two parts can trade against each other. When a ball falls, it loses height and gains speed, so $U$ drops while $K$ rises. The question is whether the total stays the same. Intuition suggests it should for an ideal frictionless drop, and the rest of this article explains precisely when that holds.

Conservative and Non-Conservative Forces

The conservation statement depends on the kind of force doing work. A conservative force is one whose work depends only on the starting and ending positions of an object, not on the path taken between them. Gravity near the Earth's surface and the elastic force of an ideal spring are the standard examples. If you lift a book from a table to a shelf by a straight route or a zigzag route, gravity does the same amount of work either way. Because of this path independence, it is possible to assign a potential energy function to a conservative force, so that the work it does equals a decrease in $U$.

A non-conservative force does not have this property. Kinetic friction is the classic case: the longer the path an object slides, the more work friction removes from its motion, so the work depends on the path. Air drag, rolling resistance and any applied push or pull whose work is not recovered behave the same way. Non-conservative forces cannot be attached to a potential energy function, and their work leaves a lasting change in the mechanical energy of the system.

$\text{Conservative: } W_{\text{path}} \text{ depends only on endpoints}$

The Conservation Statement

The principle is now simple to state. If only conservative forces do work on a system, the total mechanical energy is constant:

$K_1 + U_1 = K_2 + U_2$

Here the subscript 1 refers to any initial state and 2 to any later state. The equation does not tell us how the motion unfolds in between, which is precisely its power. It connects two states directly, bypassing the time evolution entirely. The derivation of this result from the work-energy theorem is covered in its own article, so we accept it here as the core rule.

When non-conservative forces also act, the statement must be modified. Their net work is added to the energy balance, so the total mechanical energy changes by exactly the amount of that work. The conservation law therefore remains the baseline: it is exact in the ideal case and tells us where the energy goes when it is not.

The following flow summarises how to decide whether the simple form applies.

Pendulums, Slides and Springs

The three standard systems show the principle at work. In a pendulum, the bob's gravitational potential energy is greatest at the highest points of its swing, where it momentarily stops, and greatest kinetic energy occurs at the lowest point. If the string tension does no work, which it does not because it is always perpendicular to the motion, and air drag is negligible, the bob returns to its original height at each end of the swing. For a bob released from rest at height $h$ above its lowest point, the speed at the bottom satisfies $\tfrac12 mv^2 = mgh$, so the speed depends only on the drop height, not on the mass or the string's angle.

A frictionless slide works identically. An object released from rest at height $h$ reaches the bottom with speed $v = \sqrt{2gh}$, whatever the curved shape of the track, provided the normal force does no work, which holds because it is perpendicular to the displacement. The shape of the path changes how long the trip takes, but not the final speed.

A spring stores energy as $U = \tfrac12 kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium. A block pushed against a spring and released converts this stored energy into kinetic energy, so that at the equilibrium point all the energy is kinetic. The energy conservation approach gives the block's speed at equilibrium directly, without needing to know the time it takes.

Friction as Energy Loss

Friction is the most common reason that mechanical energy is not conserved in practice. When a block slides across a rough surface, kinetic friction does negative work on it, removing mechanical energy from the block's kinetic and potential store. The energy does not vanish. It appears as thermal energy in the block and the surface, and as sound and deformation, so the total energy of the universe is conserved while the mechanical part decreases. This is why a rolling ball eventually stops and a swinging pendulum gradually shrinks its arc.

In the energy balance, friction is treated as a loss term. If $W_{\text{friction}}$ is the work done by friction (a negative number), then the mechanical energy at the end is smaller than at the start by an amount equal to the magnitude of that work. The precise treatment of friction work along a path is developed in the articles on variable forces and the work-energy theorem, so the important idea here is qualitative: friction is the signal that the simple conservation rule must be corrected by an explicit loss.

The chart below illustrates the bookkeeping for a block sliding down a rough incline, showing how the starting energy divides between what remains mechanical and what is lost to heat.