CLASSICAL MECHANICS
Specificity of Newton's laws for objects with variable mass

Basic terms

Dynamics (mechanics)

In mathematics and physics, dynamics is the branch of mechanics that is concerned with the effects of forcess on the motion of objects.

Force

Force isn't really a fundamental quantity in physics, despite the inertia of physics education still introducing students to physics via the Newtonian concept of force. More fundamental are momenta, energy and stress. In fact, no one measures force directly. Instead, everytime one says one is measuring a force, a quick rethinking would make one realize that what one really measures is stress (or maybe its gradient). The "force" you feel on your skin, for example, is really your pressure nerve cells picking up a change in pressure. A spring meter measures the tension of the spring, which is really its stress, etc. etc.

In physics, a net force acting on a body causes that body to accelerate (i.e. to change its velocity). Force is a vector. The SI unit used to measure force is the newton.

Force was first described by Archimedes. The total (Newtonian) force on a point particle at a certain instant in a specified situation is defined as the rate of change of its momentum

∑F = dp/dt = d(mv)/dt

Where m is the inertial mass of the particle, vo is its initial velocity, v is its final velocity, and T is the time from the initial state to the final state; the expression on the right of the equation being the limit as T goes to zero.

Force was so defined in order that its reification would explain the effects of superimposing situations: If in one situation, a force is experienced by a particle, and if in another situation another force is experience by that particle, then in a third situation, which (according to standard physical practice) is taken to be a combination of the two individual situations, the force experienced by the particle will be the vector sum of the individual forces experienced in the first two situations. This superposition of forces, along with the definition of inertial frames and inertial mass, are the empirical content of Newton's laws of motion.

Gravity

Gravitation is the force of attraction that exists between all particles with mass in the universe. It is the force of gravity which is responsible for holding objects onto the surface of planets and, with Newton's law of inertia is responsible for keeping objects in orbit around one another.

"Gravity is the force that pulls you down." -- Merlin in Disney's The Sword in the Stone

Merlin was right, of course, but gravity does much more than just hold you in your chair. It was the genius of Isaac Newton to recognize that. Newton recalled in a late memoir that while he was trying to figure out what kept the Moon in the sky, he saw an apple fall to the ground in his orchard, and he realized that the Moon was not suspended in the sky, but continuously falling, like a cannon ball that was shot so fast that it continuously misses the ground as it falls away due to the curvature of the Earth.

If one wishes to be precise, one should distinguish between gravitation, the universal force of attraction, and gravity, which is the resultant, on the Earth's surface, of the attraction by the earth's masses, and the centrifugal pseudo-force caused by the Earth's rotation. In casual discussion, gravity and gravitation are often used interchangeably.

By Newton's third law, any two objects exert equal and oppositely directed gravitational pull on each other.

Kinetic energy

In physics, kinetic energy is energy possessed by a body by virtue of its motion. In Newtonian mechanics, a body with mass m, moving in a straight line with velocity v, has a translational kinetic energy of

T = m v²/2.

If a body is rotating, its rotational kinetic energy equals

T = I ω²/2.

where I is its moment of inertia and ω its angular velocity.

Where gravity is weak, and objects move at much slower velocities than light (e.g. in everyday phenomena on Earth), Newton's formula is an excellent approximation of relativistic kinetic energy.

Mass

Mass is a property of physical objects which, roughly speaking, measure the amount of matter contained in an object. It is a central concept of classical mechanics and related subjects. In the SI system of measurement, mass is measured in kilograms.

Strictly speaking, mass refers to two quantities:

• Inertial mass is a measure of an object's inertia, which is its resistance to changing its state of motion when a force is applied. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily.
• Gravitational mass is a measure of the strength of an object's interaction with the gravitational force. Within the same gravitational field, an object with a smaller gravitational mass experiences a smaller force than an object with a larger gravitational mass. (This quantity is sometimes confused with weight.)

Inertial and gravitational mass have been experimentally shown to be equivalent, as accurately as we can measure, although they are conceptually quite distinct. Below, we will discuss the definitions and implications of each of these two quantities.

Momentum

Momentum is the Noether charge of translational invariance. As such, even fields as well as other things can have momentum, not just particles. However, in curved spacetime which isn't asymptotically Minkowski, momentum isn't defined at all.

In physics, momentum is a physical quantity related to the velocity and mass of an object.

In classical mechanics, momentum (traditionally written as p) is defined as the product of mass and velocity. It is thus a vector quantity.

The SI unit of momentum is newton-seconds, which can alternatively be expressed with the units kg.m/s.

Rocket

A rocket is a vehicle, missile, aircraft (or the engine employed to propel these) which operates by ejecting a reaction mass formed by the combustion of a propellant with an oxidiser, both of which are carried by the rocket; see Newton's 3rd Law of Motion. Rockets range in size from tiny models that can be purchased at a hobby store, to the enormous Saturn V used for the Apollo program.

Rockets are commonly used when it is necessary to carry all the fuel a vehicle needs (such as in outer space) and there is no other substance (land, water, or air) that a vehicle may push itself with. There are many different types of rockets, and a comprehensive list can be found in spacecraft propulsion.

Most current rockets are chemical rockets. A chemical rocket engine may use solid fuel, like the Space Shuttle's SRBs, or liquid fuel, like the Space Shuttle's main engines. A chemical reaction is initiated with the fuel in the combustion chamber, and the hot gases are forced out of a nozzle (or nozzles) at the back end of the rocket. The jet of gases generates thrust that propels the rocket forward.

Another class of rockets in increasingly common use are ion thrusters, which use electrical rather than chemical energy to accelerate their reaction mass. Nuclear thermal rockets have also been developed, but never put into use.

Rockets were first developed by the Chinese as early as B.C. 300, using gunpowder. These were initially developed for entertainment, the precursors to modern fireworks, but were later adapted for warfare in the 11th century. Because the pressures on the rocket walls are lower, the use of rockets in warfare preceded the use of the gun, which required a higher level of metal technology. It was in this role that rockets first became known to Europeans following their use by Ottomans at the siege of Constantinople in 1453. For several more centuries they remained curiosities to those in the West.

Space

The definition of space in physics is contentious. Various concepts used to try to define space have included:

• the structure defined by the set of "spatial relationships" between objects
• a manifold defined by a coordinate system where an object can be located.
• the entity that stops all objects in the universe from touching one another

In classical physics, space is a three-dimensional Euclidean space where any position can be described using three coordinatess. Relativistic physics examines spacetime rather than space; spacetime is modeled as a four-dimensional manifold.

Statics

Statics is the branch of physics that is concerned with physical systems that are 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 rest under the action of external forces of equilibrium. When in static equilibrium, the system is either at rest or moving at constant velocity through its center of mass.

By Newton's second law, this situation implies that the net force and net torque on every subsystem is zero, meaning that for for every force bearing upon a member, there must be an equal and opposite force. From this constraint, such quantities as stress or pressure can be derived. The net forces equalling zero is known as the first condition for equilibrium, and the net torque equalling zero is known as the second condition for equilibrium.

Statics is thoroughly used in the analysis of structures, for instance in architectural engineering. Strength of materials is a related field of mechanics that relies heavily on the application of static equilibrium.

Time

One can say that one event occurs after another event. Furthermore one can measure how much one event occurs after another. The answer to how much is the amount of time between the those two events.

One way of defining the idea of 'after' is based on the assumption of causality. The work humanity has done to increasingly understand the nature and measurement of time, through the work of making and improving calendars and clocks, has been a major engine of scientific discovery.

Vector (spatial)

The concept of a vector is fundamental in physics and engineering. Although the word now has many meanings (see also vector, and generalizations below), its original and most common meaning in those fields is a quantity that has a close relationship to spatial directions. The use of vector in this article refers to that original meaning, except where otherwise noted.

Often informally described as an object with a "magnitude" (size) and "direction", a vector is more formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry.

Such a vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus.