created at January 30, 2021

# Lesson 1 - About mechanics

## Introduction

The whole our universe is filled with matter and various objects. Let's call them - bodies. All these bodies (and we people as well) are interacting with each other somehow. Generally, our world is very complex and we should accept the fact, that we know not much about it. However, things usually happen according to some patterns or rules. It is obviously very useful for us to know and understand such rules since they let us to predict behavior of physical systems and control them. Since as such is exactly concerned with studying and exploring such rules (or better - laws). In particular, "physics" explores the material interaction between bodies. This interaction can have different nature - mechanical, electromagnetical etc.

## Theoretical Mechanics

Mechanics is that area of science concerned with mechanical interaction between bodies. It explores the laws of body motion with the forces applied to them. You can feel this interaction everywhere - gravitation, driving a car, the river flow, asteroid and planet collisions, a hurricane over the rooftop of your house. In other words, mechanics is everywhere and that's why it's essential to figure out the basic principles of mechanics. This is a powerful asset in your education. Since you understand basics, you will be able to find the answers for most questions you are curious about - why the Leaning tower of Pisa is still standing, why it's better to change winter tires seasonably, and why it's better to release gas pedal while skidding with a rear-wheel drive vehicle (if only you are not a drifter).

## Models

One of the most essential tools science operates with is a Model. The Model itself is some entity, which imitates the behavior of some real object. The model has similarity with the target object. This lets us to investigate the nature of the real object by substituting it with the model, when use of the real object is difficult or even impossible. Consider, you have decided to build an airplane, but you can't predict it's behavior while moving in real air flow. Thus, you create a small copy of the airplane and put it inside the wind tunnel, which allows you to explore models aerodynamical parameters. Build the real airplane for such a purpose is not safe and impractical. Another case is when you decided to build a breadboard model of the building in order to share it's details with your colleagues during meeting. All these are referred as physical models.

One of the most important things while building the model is idealization of the target system. You should to figure out the most essential aspects of your system and to implement them in the model. The model shouldn't necessarily to be an exact copy of the system, it is designed just to reproduce some aspect of the system behavior, which is interesting to us. For instance, when we create a small copy of an airplane for wind-tunnel investigations, the only thing we are concerned with is a geometrical shape of the model - it should be similar. Neither the material or color, nor the weight of the model is essential for us.

The mathematical model is a description of the real object by the language of mathematics. It is a set of equations, inequalities, etc., which describes the nature of the object. Since the real world objects are very complex - the attempt to describe an object in absolute precision would lead to enormous amount of equations. We need to idealize the system - to neglect all that has minor effects and to concerned with the most essential things. That's the greatest ability of the talented scientist or engineer. Of course, we introduce some error to the model together with neglecting some secondary factors, but this leads to a simple and useful model, and it's worth it.

You basically are dealing with mathematical models on physics lessons in school. When you are performing a laboratory work - you are dealing with physical models like pendulum, breadboard model of the electrical circuit, test tubes with chemicals etc.

There is a set of common mathematical models, widely used in theoretical mechanics, like "material point", "rigid body", "continuous medium".

Theoretical mechanics is based on axioms and theorems. Axiom is a statement, which doesn't require proof. It is being trusted because of plenty years of experience and practical observations. Theorems are obtained from axioms by using strictly logical ways. Thus, theoretical mechanics is a fundamental science.

## Applicability

Another thing to mention is applicability of theoretical mechanics. It gives correct results for bodies, which sizes are much greater than particles these bodies consist of (atoms). If this is not the case - we have to deal with quantum effects, and this is not the field of our conversation. Also the velocity of body motion should be much less than speed of light (299 792 458 m/s). Otherwise, relativistic mechanics takes place. It's clear that for almost all cases we have to deal with in our everyday life these conditions are fulfilled.

The presentation of theoretical mechanics is traditionally split into three parts: statics, kinematics, and dynamics. First you learn statics. It gives you understanding of such basic and very important things as forces, reactions, mechanical system equilibrium conditions. No motion happens here. Next you learn kinematics. Here you discover basic parameters that describes body motion (velocity, angular velocity, acceleration, trajectory). However, at this stage it's not clear what caused this motion. The third part: dynamics - links statics and kinematics together, and gives you an answers on how will move bodies under applied forces.