Motion - mechanics

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MOTION MECHANICS

There are 2 types of mechanics in physics- classical mechanics and quantum mechanics- as this is a high school physics guide, the focus shall be mainly on classical mechanics seeing as quantum mechanics is highly complicated and tricky in places. Classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology.

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics. Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other major sub-field of mechanics, quantum mechanics, which reconciles the macroscopic laws of physics with the atomic nature of matter and handles the wave-particle duality of atoms and molecules. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. General relativity unifies special relativity with Newton's law of universal gravitation, allowing physicists to handle gravitation at a deeper level.

Statics

Statics is the branch of mechanics concerned with the analysis of loads (force, torque/moment) on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity. When in static equilibrium, the system is either at rest, or its center of mass moves at constant velocity.

By Newton's first law, this situation implies that the net force and net torque (also known as moment of force) on every body in the system is zero. From this constraint, such quantities as stress or pressure can be derived. The net forces equaling zero is known as the first condition for equilibrium, and the net torque equaling zero is known as the second condition for equilibrium.

Vectors

A scalar is a quantity like mass or temperature which only has a magnitude. A vector is a quantity that has both a magnitude and a direction. There are many notations to identify a vector, the most common ones are:

A bold faced character V
An underlined character V
A character with an arrow over it ${\overrightarrow {V}}$ .

Vectors can be added using the parallelogram law or the triangle law . Vectors contain components in orthogonal bases. Unit vectors i, j, and k are along the x, y, and z directions.

Force

Force is the action of one body on another. A force tends to move a body in the direction of its action. The action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action.

Forces are classified as either contact or body forces. A contact force is produced by direct physical contact; an example is the force exerted on a body by a supporting surface. On the other hand, a body force is generated by virtue of the position of a body within a force field such as a gravitational, electric, or magnetic field. An example of a body force is your weight.

Moment of a Force

In addition to the tendency to move a body in the direction of its application, a force can also tend to rotate a body about an axis. The axis may be any line which neither intersects nor is parallel to the line of action of the force. This rotational tendency is known as the moment M of the force. Moment is also referred to as torque.

Moment About a Point

The magnitude of the moment of a force at a point O, is equal to the perpendicular distance from O to the line of action of F, multiplied by the magnitude of the force. Simply the magnitude of the moment is defined as

$M=Fd$

where F is the force applied (N) and d is the perpendicular distance from the axis of the line of action to the force (m).

Moment of Inertia

In classical mechanics, moment of inertia, also called mass moment, rotational inertia, polar moment of inertia of mass, or the angular mass, (SI units kg·m²) is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.

While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.

Dynamics

In classical mechanics, analytical dynamics, or more briefly dynamics, is concerned about the relationship between motion of bodies and its causes, namely the forces acting on the bodies and the properties of the bodies (particularly mass and moment of inertia). The foundation of modern day dynamics is Newtonian mechanics and its reformulation as Lagrangian mechanics and Hamiltonian mechanics.

Newton's Law of Motion

Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces. They have been expressed in several different ways over nearly three centuries, and can be summarized as follows:

First Law: The velocity of a body remains constant unless the body is acted upon by an external force.
Second Law: The rate of change of momentum of an object is directly proportional to the resultant force acting upon it, this can also be written mathematically,

$F={\frac {dp}{dt}}=m{\frac {dv}{dt}}+v{\frac {dm}{dt}}$

for a constant mass system

$v{\frac {dm}{dt}}=0$

and so

$F=m{\frac {dv}{dt}}$

where ${\frac {dv}{dt}}=a$

reducing to

$F=ma$

3. Third Law: The mutual forces of action and reaction between two bodies are equal, opposite and collinear, which can be written mathematically as

$\displaystyle \sum F_{a,b}=-\displaystyle \sum F_{b,a}$

Validity of Newton's Laws

Newton's Laws of Motion are incredibly useful mechanisms and explanations for the motion of macroscopic objects at relatively small velocities in relatively low intensity gravitational fields, however, at incredibly high speeds (usually involving sub-atomic particles) or in an incredibly strong gravitational field Newton's Laws of Motion do not work properly as, due to Einstein's Theory of Special Relativity, objects moving at incredibly high speeds do not have a constant mass:

$v{\frac {dm}{dt}}\neq 0,v\to 3\times 10^{8}ms^{-1}$