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# Further Mechanics and Thermal Physics

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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).

Figure 1. A pendulum swinging away from its stability position. Source: StudySmarter.

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.

Figure 2. A diagram of the molecules composing a gas. Source: StudySmarter.

## Further Mechanics and Thermal Physics - Key takeaways

• Periodic motion is of great relevance in physics. It is usually found in circular motions and in systems known as harmonic oscillators.

• Thermal physics is the branch of physics studying processes involving temperature.
• There are phenomenological aspects in thermal physics that lead to concepts such as specific heat capacity, latent heat, or the laws of ideal gases.
• There are more fundamental aspects in thermal physics leading to the modelling of substances through the kinetic model theory, which allows extracting macroscopic properties as statistical features of many microscopic phenomena.

xi : x1, x2, x3, ...

xi : the initial x value

By random motion, we mean that the movement of an object is affected by many factors and other bodies so that it is essentially impossible for us to accurately predict its motion.

Energy and momentum.

Specific heat capacity is the energy needed per unit of mass to raise the temperature of a substance by one degree Celsius.

Latent heat is the energy per unit of mass needed to change the phase of a substance.

The specific heat capacity of water has an approximate value of 4.18 [J/kg*ºC].

## Final Further Mechanics and Thermal Physics Quiz

Question

The behaviour of systems around a stable point is that of a harmonic oscillator.

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In thermal physics, we find phenomenological and fundamental aspects.

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Charle’s law relates volume and temperature for fixed pressure.

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The specific heat capacity of a substance is measured as temperature per unit of mass.

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Macroscopic properties in kinetic molecular theory are statistical features extracted from microscopic models.

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Can we consider electromagnetic waves as having a periodic motion?

Yes, we usually characterise them with frequency, which is a characteristic of periodicity.

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What are the two factors that contribute to the energy of a harmonic oscillator?

Amplitude and frequency.

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What is the name of the state around which a harmonic oscillator oscillates?

The stable state or stability position.

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What are the two main aspects of thermal physics?

Phenomenological aspects and fundamental aspects.

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What is specific heat capacity?

It is the energy needed per unit of mass to raise the temperature of a substance by one degree.

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What is latent heat?

Latent heat is the energy per unit of mass needed to change the phase of a substance.

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What are the laws of ideal gases?

Charle’s law, Boyle’s law and Gay-Lussac’s law.

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What are the assumptions of the model of ideal gases?

Point-like particles, constant movement, and elastic collisions.

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What is the definition of pressure?

Pressure is the average force exerted per unit of surface by the particles of a substance on the boundary of the region in which the substance is contained.

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What is the definition of temperature?

It is a measure of the average kinetic energy of a substance’s particles.

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What is the definition of a radian?

It is the angle subtended at a circle’s centre by an arc of equal length to the radius of a circle.

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What is the name of the force that induces circular motion?

Centripetal force.

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What is the direction of the velocity in circular motion compared to the circular path?

It is perpendicular to the circular path.

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What is the centrifugal force?

It is a pseudo force felt by an object in circular motion.

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What units do we use for the time period in circular motion?

Seconds.

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What units do we use for frequency in circular motion?

Hertz.

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What is the friction force equal to in a car moving along a curve?

To the centripetal force.

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What units do we use for angular speed?

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What is the angular speed of an object if it takes 12 seconds to complete an orbit?

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What is the angular speed of an object if it takes 3 seconds to complete three orbits?

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If an object A takes less time to complete an orbit than an object B, which one has a larger angular velocity?

Object A.

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Can angular displacement be larger than one orbit (2π or 360degrees)?

Yes, it can.

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The centripetal force exerted on a ball of 1.5kg moving in circular motion is equal to 122 Newtons. Calculate the acceleration of the ball. Then, knowing that the ball is moving 3 metres from the centre of rotation, determine its velocity.

v = 15.62m/s.

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Which of the following is the symbol for period in seconds?

T.

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Which of the following is the name given to the number of times an event happens in one unit time?

Frequency.

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While the circular motion is present, what happens to the object’s distance from the axis of rotation?

It stay constant at all times.

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Which of the following is not one of the types of periodic motion?

Linear motion.

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A tennis ball connected to a stick with a 1.5m long rope is following a circular motion around the stick at a constant velocity. The tennis ball has a mass of 70g and is moving at a velocity of 18.4 m/s. Calculate the centripetal force.

15.8 [N].

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A cube with a mass of 6kg is attached to a spring with a force constant of 4 N/m. Calculate the period and the frequency of the cube if a force of 50N is applied to compress the string.

7.7 [s].

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Find the acceleration due to gravity for a simple pendulum with a length of 90cm and a period of 2.5246s.

5.57 m/s².

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For small displacements, a pendulum can be considered a simple harmonic oscillator. True or false?

True.

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What is the symbol for the force constant?

k.

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Periodic motion is a motion that repeats itself at certain time intervals. True or false?

True.

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The velocity of an object in a circular motion stays constant. True or false?

False.

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Which law is used to explain simple harmonic motion?

Hooke’s law.

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What is thermal energy transfer?

It is the energy being transferred in the form of heat between bodies.

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Which of the following is not a method for thermal energy transfer?

Absorption

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What is convective heat transfer?

It is the transfer of thermal energy through a fluid due to movement of the particles.

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It is thermal energy emitted from heated bodies as EM radiation.

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What is conductive heat transfer?

It is the transfer of thermal energy through collisions of subatomic particles of a body.

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What is the heat transfer coefficient?

It is a parameter that defines how well a medium can transfer thermal energy.

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What is the formula for heat transfer?

It is defined as Q= hA(T2-T1).

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How do we find the heat transfer coefficient?

By rearranging the above equation h = Q/A(T2-T1).

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Which of the following parameters does not affect the heat transfer coefficient

Thermal resistance

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Which of the following parameters does not affect heat flow?

Volume

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