2.3.4.5. Energy Dissipation

Table of Contents

Where Mechanical Energy Goes

In many real situations, mechanical energy is not fully conserved. A moving object slows down, a bouncing ball does not return to its original height, and a sliding block becomes warm. In these cases, energy is not destroyed. Instead, some of the mechanical energy is transformed into other forms, often thermal energy, sound, deformation, or internal energy of the objects involved. This transformation is called energy dissipation.

Energy dissipation is especially important when non-conservative forces act, such as friction, air resistance, or internal material forces during collisions and deformations. The total energy of the universe is still conserved, but the easily tracked mechanical energy, which is the sum of kinetic and potential energy, decreases.

Mechanical energy can decrease in the presence of dissipative forces, but total energy is always conserved.

Dissipative Processes

A dissipative process is one in which organized motion changes into less organized microscopic motion. For example, when a box slides across a rough floor, its motion becomes slower, while the surfaces in contact heat up slightly. The lost kinetic energy has been transferred into internal energy.

Common sources of energy dissipation are shown below.

Dissipative effect	What happens to mechanical energy
Friction	Converted mostly into thermal energy
Air resistance	Converted into thermal energy of air and object, sometimes sound
Inelastic deformation	Converted into internal energy and permanent shape change
Sound production	Converted into wave energy in the surrounding medium
Vibrations in materials	Converted gradually into heat

A useful way to think about dissipation is that energy spreads into many tiny microscopic motions that are difficult to recover as useful macroscopic motion.

Friction as a Cause of Dissipation

When friction acts, it usually removes mechanical energy from the visible motion of a system. For a block sliding on a horizontal surface, friction opposes the displacement, so the work done by friction is negative.

If a kinetic friction force $f_k$ acts over a distance $d$, then the work done by friction is

$$
W_{\text{fric}} = -f_k d
$$

This negative work represents the decrease in mechanical energy of the system, if no other non-conservative work is added.

For kinetic friction,

$$
f_k = \mu_k N
$$

so on a horizontal surface, where $N = mg$,

$$
W_{\text{fric}} = -\mu_k mgd
$$

The work done by friction is usually negative because friction opposes the direction of motion.

Energy Accounting with Dissipation

When dissipation is present, the simple conservation relation for mechanical energy must be modified. Instead of saying that kinetic plus potential energy stays constant, we account for the energy transferred away from mechanical forms.

A common relation is

$$
K_i + U_i + W_{\text{nc}} = K_f + U_f
$$

where $W_{\text{nc}}$ is the work done by non-conservative forces.

If the non-conservative force is friction, then $W_{\text{nc}} < 0$, so the final mechanical energy is less than the initial mechanical energy.

Another useful form is

$$
E_{\text{mech},i} = E_{\text{mech},f} + E_{\text{diss}}
$$

where $E_{\text{diss}}$ is the amount of energy transformed into non-mechanical forms.

A decrease in mechanical energy does not mean energy has vanished. It means energy has been transferred to forms such as heat, sound, or internal deformation.

Thermal Energy and Internal Energy

In beginner mechanics, dissipation often appears as thermal energy. At the microscopic level, atoms and molecules jiggle more vigorously when temperature rises. Friction and impact can increase this microscopic motion.

Suppose a sled slows down while moving over snow. Its kinetic energy decreases. At the same time, the sled runners and snow may warm slightly. This warming corresponds to an increase in internal energy.

Internal energy is a broader idea than thermal energy alone. It can include molecular motion, molecular vibration, and energy stored in deformed materials. In dissipative situations, mechanical energy often becomes internal energy.

Inelastic Collisions and Dissipation

A collision is called inelastic when kinetic energy is not conserved. During such a collision, some kinetic energy is transformed into sound, heat, or deformation.

For example, when a lump of clay hits the floor and sticks, the total momentum may still be conserved under the right conditions, but the kinetic energy after impact is smaller than before. The missing kinetic energy has been dissipated.

This does not violate energy conservation. It only means that the energy is no longer entirely in the form of macroscopic motion.

Everyday Examples

A car braking to a stop is a familiar example of energy dissipation. The car’s kinetic energy is converted mainly into thermal energy in the brakes, tires, and road. A bouncing ball also shows dissipation. Each bounce is lower because some mechanical energy is transformed into sound, heat, and internal deformation during impact.

A pendulum swinging in air gradually slows down because of air resistance and friction at the pivot. The energy is slowly dissipated into the air and the support.

Visualizing Dissipation

The diagram below shows a block sliding on a rough surface. As it moves, friction acts opposite the motion and mechanical energy decreases.

Rendering scientific sketch…

Block sliding with friction

Because the friction force points opposite the displacement, it does negative work, reducing the mechanical energy of the block.

A Simple Example

Consider a block of mass $m$ sliding with initial speed $v_i$ across a rough horizontal floor and coming to rest after traveling distance $d$.

Initially, its kinetic energy is

$$
K_i = \frac{1}{2}mv_i^2
$$

Finally, since it stops,

$$
K_f = 0
$$

If friction is the only non-conservative force, then

$$
W_{\text{fric}} = K_f - K_i
$$

$$
-f_k d = 0 - \frac{1}{2}mv_i^2
$$

and therefore

$$
f_k d = \frac{1}{2}mv_i^2
$$

This shows that the initial kinetic energy was dissipated by friction.

Interpreting Lost Energy Correctly

In physics, people often say energy is "lost" to friction. This phrase is convenient, but it can be misleading. The energy is not truly lost. It has become less useful for producing large-scale motion because it has been spread into microscopic forms.

For this reason, dissipated energy is sometimes described as degraded energy. It still exists, but it is harder to recover completely as organized mechanical energy.

“Lost energy” in mechanics usually means “energy transformed into non-mechanical forms,” not energy destroyed.

Summary Relation

In dissipative systems, the key idea is that mechanical energy decreases while total energy remains constant. A compact statement is

$$
\Delta E_{\text{mech}} = -E_{\text{diss}}
$$

if no external energy is added to the system.

This chapter’s main message is simple. Friction, drag, and inelastic effects do not break energy conservation. They redirect energy from kinetic and potential forms into internal energy, heat, sound, and deformation.

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