Thermal energy refers to the energy contained within a system that is responsible for its temperature. Heat is the flow of thermal energy. A whole branch of physics, thermodynamics, deals with how heat is transferred between different systems and how work is done in the process (see the 1ˢᵗ law of thermodynamics).

In the context of mechanics problems, we are usually interested in the role thermal energy plays in ensuring conservation of energy. Almost every transfer of energy that takes place in real-world physical systems does so with efficiency less than 100% and results in some thermal energy. This energy is usually in the form of low-level thermal energy. Here, low-level means that the temperature associated with the thermal energy is close to that of the environment. It is only possible to extract work when there is a temperature difference, so low-level thermal energy represents 'the end of the road' of energy transfer. No further useful work is possible; the energy is now 'lost to the environment'.

Consider the example of a man pushing a box across a rough floor at a constant velocity as shown in Figure 1. Since the friction force is non-conservative, the work done is not stored as potential energy. All the work done by the friction force results in a transfer of energy into thermal energy of the box-floor system. This thermal energy flows as heat within the box and floor, ultimately raising the temperature of both of these objects.

Finding the change in total thermal energy $\mathrm{\Delta }{E}_T$‍ of the box-floor system can be done by finding the total work done by friction as the person pushes the box. Recall that the box is moving at constant velocity; this means that the force of friction and the applied force are equal in magnitude. The work done by both these forces is therefore also equal.

Using the definition of work done by a force parallel to the motion of an object moving through a distance $d$‍:

If the coefficient of kinetic friction is ${\mu }_k$‍ then this can also be written as

Exercise 1a: Suppose the person shown in Figure 1 pushes the box, maintaining a constant velocity. The box has a mass of $100\mathrm{kg}$‍ and moves through a distance of $100\mathrm{m}$‍. The coefficient of kinetic friction between the box and floor is ${\mu }_k=0.3$‍. How much thermal energy will be transferred to the box-floor system?

Exercise 1b: When the person pushes on the box, they rely on friction between the soles of their shoes and the floor. Is there any change in the thermal energy of the persons shoes due to pushing on the box?

The force of drag on a moving object due to a fluid such as air or water is another example of a non-conservative force.

When an object moves through a fluid, some momentum is transferred and the fluid is set in motion. If the object were to stop moving there would still be some residual motion of the fluid. This would die down after some time. What is happening here is that the large scale motions of the fluid are eventually re-distributed into many smaller random motions of the molecules in the fluid. These motions represent an increased thermal energy in the system.

Figure 2 shows a system in which a thermally insulated water tank has a shaft suspended in it. Two paddles are attached to the shaft which is set to rotate on its axis. In this system, any work done in rotating the shaft results in a transfer of kinetic energy to the water. If the drive force is removed from the shaft after some time, there will still be some residual motion. However, the motion will eventually die down and result in an increase in thermal energy of the water.

Interestingly, a system similar to that shown in Figure 2 was used by James Prescott Joule (1818 – 1889), for whom the SI unit of energy is named. Using a paddle wheel submerged in a tank of whale oil and driven by falling weights he was able to determine the relationship between mechanical energy and heat. This lead to the law of conservation of energy and the 1ˢᵗ law of thermodynamics.

Exercise 2a: Suppose the paddle wheel depicted in Figure 2 is rotated by an electric motor which is rated at 10 W output power for 30 minutes. How much thermal energy is transferred to the water?

Exercise 2b (extension): If the tank initially contains $1\mathrm{L}$‍ of water at ${10}^{\circ }\mathrm{C}$‍ then what would be the water temperature after the motor is stopped and the water stops sloshing around?