Further Mechanics and Thermal Physics

The reason to study further mechanics and thermal physics is to understand advanced topics, such as the harmonic oscillator or the significance of thermodynamic properties in models such as the ideal gas. These simple systems are widely used in physics to describe a myriad of phenomena, which is why a good understanding is essential.

What is periodic motion?

Periodic motion is a type of motion that repeats itself after a certain time interval. These motions are of special relevance in physics because they occur in many day-to-day phenomena, such as in electromagnetic waves or when we enjoy ourselves on a swing.

Periodic motion represents a very good approximation of a stable system under a small perturbation. Picture a marble in a bowl: its stability is achieved by staying still in the centre of the bowl, but any displacement from that centre will cause it to oscillate and eventually fall back into the centre due to friction. Any stable system (in this case, the centre of the bowl) follows a periodic evolution when slightly disturbed.

Circular motion

The motion followed by an object whose distance to a certain point called centre is constant is called circular motion. It is a periodic motion because the body will eventually return to the position it previously occupied. Usually, we are interested in bodies moving in circles at a uniform rate, which yields a time periodicity.

Harmonic oscillator

Whereas circular motion is not necessarily periodic (although it usually is), harmonic oscillators are defined precisely by their periodic behaviour. They are systems that oscillate with a certain period around a stable position.

The best example of a harmonic oscillator is a pendulum on earth with no air resistance. The stability position is the string of the pendulum being vertical. When moved from that position, the pendulum will oscillate from one side to the other.

There are many characterisations and quantities we can assign to these systems. Here, the point to be made is that bodies following such a movement have an energy that is characterised by two main factors: the amplitude of the movement (how far its oscillations take it away from the stability position) and the frequency of the oscillation (the rate of completed oscillations per unit of time).

As a last relevant feature of harmonic motion, we are going to define the concept of resonance. It turns out that different harmonic motions interfere with each other. A mathematical description reveals that they ‘add up’ and can either neutralise each other, yield a mixed result, or interfere constructively to amplify the oscillations.

When a constructive interference occurs, the amplification can be notable and maximise the transfer of energy between the harmonic oscillations. This phenomenon of optimal transfer of energy is called resonance. A radio constitutes a day-to-day example of this: radio waves are being transmitted all across the world. While we cannot hear them, our radios are devices where other waves are tuned to constructively interfere with these radio waves, thus amplifying their energy and generating a sound.

What is thermal physics?

Thermal physics is the study of physical processes involving temperature changes. Here, we are going to look at some of its phenomenological and fundamental aspects.

Phenomenological aspects of thermal physics

The phenomenological aspects of thermal physics include different laws and processes proposed as part of theories based solely on the realisation of experiments. Two examples are thermal energy transfer experiments and ideal gases.

Thermal energy transfer is the process by which bodies with different temperatures transfer their energy from one to another, thus changing their temperatures. In thermal energy transfer experiments, the following are widely studied:

• Specific heat capacity, which refers to the amount of heat (thermal energy) needed to raise the temperature of 1 kg of a certain substance by 1 degree Celsius. Specific heat capacity is a specific property in that it is different for each substance.
• Latent heat, which is the heat needed to change the phase of 1 kg of a substance. By this change of phase, we mean the process by which substances change between their gas, liquid, and solid states. In this process, the temperature does not change (boiling water, for instance, remains at 100ºC until it is completely vaporised), so the characterisation does not include a change of temperature.

Ideal gases are gases that exist under certain conditions and follow very simple laws while varying with regard to some properties (see below under ‘Fundamental aspects’ for some of their basic characteristics). Three processes involving ideal gases that concern their temperature are the following:

• When a certain amount of an ideal gas is held at a constant temperature, changes in pressure are inversely proportional to changes in volume. This is known as Boyle’s law.
• When a certain amount of an ideal gas is held at constant pressure, temperature changes are directly proportional to changes in volume. This is known as Charle’s law.
• When a certain amount of an ideal gas is held at a constant volume, changes in pressure are directly proportional to changes in temperature. This is known as Gay-Lussac’s law.

Fundamental aspects of thermal physics

This takes us to molecular kinetic theory, which extracts global properties from substances, especially gases, as collective effects of the molecules forming the substance.

In general, the molecules of a liquid substance follow a Brownian motion, which is the term used for a random motion caused by the influence of a huge number of other particles interacting with the molecules in the substance.

In a simplifying hypothesis, the particles of a substance are assumed to be point-like, in constant motion (not in solid-state), and to have elastic collisions (without loss of energy or momentum). These conditions are part of what is known as kinetic molecular theory and comprise the definition of ideal gases.

Nevertheless, whether we adopt a simplified description or study a substance without any assumptions, we are always able to extract statistical conclusions from the behaviour of all the particles forming the substance. For instance, the temperature is a measure of the mean kinetic energy of the particles: the higher the temperature, the faster particles will move on average.

Another relevant example of a statistical property is pressure. Essentially, it is the average force per unit of surface exerted by the particles on the region in which they are contained.

The laws for ideal gases can be derived by taking into account the assumptions we mentioned and by applying Newton’s laws of motion at a microscopic level, which yields average quantities (pressure, temperature) and relations between them.

x_i : x₁, x₂, x₃, ...

x_i : the initial x value