Mechanics

From IITJEEWiki

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains, and is one of the oldest and largest subjects in science, engineering and technology.

Besides this, many related specialties exist, dealing with gases, liquids, and solids, and so on. Classical mechanics is enhanced by special relativity for objects moving with high velocity, approaching the speed of light; general relativity is employed to handle gravitation at a deeper level; and quantum mechanics handles the wave-particle duality of atoms and molecules.

Contents 1 Description of the theory 1.1 Displacement and its derivatives 2 Displacement 2.1 Velocity and speed 2.2 Acceleration 2.3 Frames of reference 2.4 Forces; Newton's Second Law 3 Limits of validity 3.1 The Newtonian approximation to special relativity

[edit] Description of the theory

We model real-world objects as point masses (negligible size). position, mass, and the forces are the basic important things that we need to know to analyze things in Mechanics.

Extremely small objects, (electron), are studied by quantum mechanics).

The center of mass of a Large object behaves like a point mass!

[edit] Displacement and its derivatives

The SI derived units with kg, m and s
displacement	m
speed	m s−1
acceleration	m s−2
jerk	m s−3
specific energy	m² s−2
moment of inertia	kg m²
momentum	kg m s−1
angular momentum	kg m² s−1
force	kg m s−2
torque	kg m² s−2
energy	kg m² s−2
power	kg m² s−3
pressure	kg m−1 s−2
surface tension	kg s−2
irradiance	kg s−3

[edit] Displacement

The displacement, or position, of a point mass is defined with respect to an arbitrary fixed reference point, O, called the origin. It is denoted as vector r from O to the particle. The point mass can move with time, so r is a function of t time.

[edit] Velocity and speed

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time or

.

Velocities are directly additive and subtractive.

For example, if one car traveling East at 60 km/h passes another car traveling East at 50 km/h, then from the perspective of the slower car, the faster car is traveling east at 60 − 50 = 10 km/h. Whereas, from the perspective of the faster car, the slower car is moving 10 km/h to the West. Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.

If the velocity of the first object is denoted by vector and the velocity of the second object by the vector where u is the speed of the first object, v is the speed of the second object, and and are unit vectors in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is:

Similarly:

When both objects are moving in the same direction (ie when vec{i}=vec{i_1}, this equation can be simplified to:

[edit] Acceleration

The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time) or

.

Acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both. If only the magnitude, v, of the velocity decreases, this is sometimes referred to as deceleration, but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.

Tangential and Normal acceleration

Component of acceleration which changes magnitude of velocity vector is called tangential acceleration. This is parallel or anti parallel to velocity vector.

Component of acceleration which changes only direction of velocity vector and is always perpendicular to velocity vector is called normal acceleration.

acosθ=

[edit] Frames of reference

While the position and velocity and acceleration of a particle can be referred to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames. They are characterized by the absence of acceleration of the observer and the requirement that all forces entering the observer's physical laws originate in identifiable sources (charges, gravitational bodies, and so forth). A non-inertial reference frame is one accelerating with respect to an inertial one, and in such a non-inertial frame a particle is subject to acceleration by fictitious forces that enter the equations of motion solely as a result of its accelerated motion, and do not originate in identifiable sources. These fictitious forces are in addition to the real forces recognized in an inertial frame. A key concept of inertial frames is the method for identifying them. (See inertial frame of reference for a discussion.) For practical purposes, reference frames that are unaccelerated with respect to the distant stars are regarded as good approximations to inertial frames.

The following consequences can be derived about the perspective of an event in two inertial reference frames, S and S', where S' is traveling at a relative velocity of to S.

• (the velocity of a particle from the perspective of S' is slower by than its velocity from the perspective of S)
• (the acceleration of a particle remains the same regardless of reference frame)
• (the force on a particle remains the same regardless of reference frame)
• the speed of light is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics have a counterpart in classical mechanics.
• the form of Maxwell's equations is not preserved across such inertial reference frames. However, in Einstein's theory of special relativity, the assumed constancy (invariance) of the vacuum speed of light alters the relationships between inertial reference frames so as to render Maxwell's equations invariant.

[edit] Forces; Newton's Second Law

Newton was the first to mathematically express the relationship between force and momentum. Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it to be a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":

.

The quantity is called the (canonical) momentum. The net force on a particle is, thus, equal to rate change of momentum of the particle with time. Since the definition of acceleration is , when the mass of the object is fixed, for example, when the mass variation with velocity found in special relativity is negligible (an implicit approximation in Newtonian mechanics), Newton's law can be written in the simplified and more familiar form

.

So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion.

[edit] Limits of validity

Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being general relativity and relativistic statistical mechanics. Geometric optics is an approximation to the quantum theory of light, and does not have a superior "classical" form.

[edit] The Newtonian approximation to special relativity

Newtonian, or non-relativistic classical momentum

is the result of the first order Taylor approximation of the relativistic expression:

so it is only valid when the velocity is much less than the speed of light. Quantitatively speaking, the approximation is good so long as