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Angular Momentum

Angular momentum

In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. In particular, if the body rotates about an axis, then the angular momentum with respect to a point on the axis is related to the mass of the object, the angular velocity and the distance of the mass to the axis. Without applying torque to the object, with respect to the reference point, the angular momentum is constant. The angular momentum is a measure for the amount of torque that has been applied over time to the object. The object has rotational inertia that resists changes in rotational motion, quantified by the moment of inertia. Angular momentum is an important concept in both physics and engineering with numerous applications. For example, the kinetic energy stored in a massive rotating object such as a flywheel is proportional to the angular momentum.

Angular momentum in classical mechanics

Definition

The traditional mathematical definition of the angular momentum of a particle about some origin is: ::

\mathbf = \mathbf \times \mathbf

:where ::L is the angular momentum of the particle, ::r is the position of the particle expressed as a displacement vector from the origin, ::p is the linear momentum of the particle, and ::

\times \,

is the cross product. Because of the cross product, L is a pseudovector perpendicular to both the radial vector r and the momentum vector p. If a system consists of several particles, the total angular momentum about an origin can be obtained by adding (or integrating) all the angular momenta of the constituent particles. Angular momentum can also be calculated by multiplying the square of the displacement 'r',the mass of the particle and the angular velocity. For many applications where one is only concerned about rotation around one axis, it is sufficient to discard the pseudovector nature of angular momentum, and treat it like a scalar where it is positive when it corresponds to a counter-clockwise rotations, and negative clockwise. To do this, just take the definition of the cross product and discard the unit vector, so that angular momentum becomes: :

L = |\mathbf||\mathbf|\sin\theta_

where θ_r,p is the angle between r and p measured from r to p; an important distinction because without it, the sign of the cross product would be meaningless. From the above, it is possible to reformulate the definition to either of the following: :

L = \pm|\mathbf||\mathbf_|

where r_{perpendicular} is called the lever arm distance to p. The easiest way to conceptualize this is to consider the lever arm distance to be the distance from the origin to the line that p travels along. With this definition, it is necessary to consider the direction of p (pointed clockwise or counter-clockwise) to figure out the sign of L. Equivalently: :

L = \pm|\mathbf||\mathbf_|

where p_{perpendicular} is the component of p that is perpendicular to r. As above, the sign is decided base on the sense of rotation. For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector: :

\mathbf= I \mathbf

where I is the moment of inertia of the object ω is the angular velocity.

Conservation of angular momentum

In analogy to Newton's second law for linear momentum, we have the following law about angular momentum: :

\frac = \boldsymbol

where τ is the net torque about the origin. This implies that angular momentum is a conserved quantity as long as there is no net torque applied to the particle. What's more, this conservation can be generalized to a system of particles under most conditions so that: :

\mathbf_ =  \mathrm \Leftrightarrow \sum \tau_ = 0

where τ_external is any torque applied to the system of particles. In orbits, the angular momentum is distributed within the spin of the planet itself, and the angular momentum of its orbit: :

\mathbf_ = \mathbf_ + \mathbf_

If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared amongst the planet and its satellite in order to be conserved. The conservation of angular momentum is used extensively in analyzing what is called central force motion. In central force motion, two bodies form an isolated system not influenced by outside forces, and the origin is placed somewhere on the line between the two bodies. Since any force the bodies exert on each other must be directed along this line, there can be no net torque, with respect to the aforementioned origin, on either body. Thus, angular momentum is conserved. Constant angular momentum is extremely useful when dealing with the orbits of planets and satellites, and also when analyzing the Bohr model of the atom!

Angular momentum in relativistic mechanics

In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum isn't conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant). For a system of point particles without any intrinsic angular momentum, it turns out to be :

\sum_i \bold_i\wedge \bold_i

(Here, the wedge product is used.).

Angular momentum in quantum mechanics

In quantum mechanics, angular momentum is defined like momentum - not as a quantity but as an operator on the wave function: :

\mathbf=\mathbf\times\mathbf

where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as :

\mathbf=-i\hbar(\mathbf\times\nabla)

where Image:del.gif is the gradient operator. This is a commonly encountered form of the angular momentum operator, though not the most general one. It has the following properties :

[L_i, L_j ] = i \hbar \epsilon_ L_k

\left[L_i, L^2 \right] = 0

and even more importantly commutes with the hamiltonian of such a chargeless and spinless particle :

\left[L_i, H \right] = 0

. Angular Momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is: ::

\ L^2 = \frac\frac\left( \sin\theta \frac\right) + \frac\frac

When solving to find eigenstates of this operator, we obtain the following ::

L^2 | l, m \rang = ^2 l(l+1) | l, m \rang

L_z | l, m \rang = \hbar m | l, m \rang

where ::

\lang \theta , \phi | l, m \rang = Y_(\theta,\phi)

are the spherical harmonics.

References

- E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN 521-09209-4 See chapter 3.
- Edmonds, A.R., Angular Momentum in Quantum Mechanics, (1957) Princeton University Press, ISBN 0-691-07912-9.
-
- Category:Physical quantity Category:Rotational symmetry ko:각운동량 ms:Momentum sudut ja:角運動量

Mass

:For other senses of this word, see mass (disambiguation). Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. Unlike weight, the mass of something stays the same regardless of location. It is a central concept of classical mechanics and related subjects. Strictly speaking, there are three different quantities called mass:
- Inertial mass is a measure of an object's inertia: 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.
- Passive gravitational mass is a measure of the strength of an object's interaction with the gravitational field. Within the same gravitational field, an object with a smaller passive gravitational mass experiences a smaller force than an object with a larger passive gravitational mass. (This force is called the weight of the object. In informal usage, the word "weight" is often used synonymously with "mass", because the strength of the gravitational field is roughly constant everywhere on the surface of the Earth. In physics, the two terms are distinct: an object will have a larger weight if it is placed in a stronger gravitational field, but its passive gravitational mass remains unchanged.)
- Active gravitational mass is a measure of the strength of the gravitational field due to a particular object. For example, the gravitational field that one experiences on the Moon is weaker than that of the Earth because the Moon has less active gravitational mass.

Introduction

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. One of the consequences of the equivalence of inertial mass and passive gravitational mass is the fact, famously demonstrated by Galileo Galilei, that objects with different masses fall at the same rate, assuming factors like air resistance are negligible. The theory of general relativity, the most accurate theory of gravitation known to physicists to date, rests on the assumption that inertial and passive gravitational mass are completely equivalent. This is known as the weak equivalence principle. Classically, active and passive gravitational mass were equivalent as a consequence of Newton's third law, but a new axiom is required in the context of relativity's reformulation of gravity and mechanics. Thus, standard general relativity also assumes the equivalence of inertial mass and active gravitational mass; this equivalence is sometimes called the strong equivalence principle. If one were to treat inertial mass m_i, passive gravitational mass m_p, and active gravitational mass m_a distinctly, Newton's law of universal gravitation would give as force on the second mass due to the first mass :

m_a_2=\frac.

Newton's third law, of reciprocal actions, shows that active and passive mass are proportional. As a result they can be defined to be equal.

Units of mass

In the SI system of units, mass is measured in kilograms (kg). Many other units of mass are also employed, such as: grams (g), metric tons, pounds, ounces, long and short tons, quintals, slugs, atomic mass units, Planck masses, solar masses, and eV/c². The eV/c² unit is based on the electron volt (eV), which is normally used as a unit of energy. However, because of the relativistic connection between (rest) mass and energy, E = mc² (see below), it is possible to use any unit of energy as a unit of mass instead. Thus, in particle physics where mass and energy are often interchanged, it is common to use not only eV/c² but even simply eV as a unit of mass (roughly 1.783 × 10^-36 kg). Because the gravitational acceleration is approximately constant on the surface of the Earth, a unit like the pound is often used to measure either mass or force (e.g. weight), although the pound is officially defined as a unit of mass. For more information on the different units of mass, see Orders of magnitude (mass).

Inertial mass

Inertial mass is the mass of an object measured by its resistance to acceleration. To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "mass" in any essential way. According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion :

F = \frac (mv)

where F is the force acting on the body and v is its velocity. For the moment, we will put aside the question of what "force acting on the body" actually means. Now, suppose that the mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. These are very reasonable assumptions for everyday objects, though, as we will see, the situation gets more complicated when we take special relativity into account. Another point to note is that, even in classical mechanics, it is sometimes useful to treat the mass of an object as changing with time. For example, the mass of a rocket decreases as the rocket fires. However, this is an approximation, based on ignoring pieces of matter which enter or leave the system. In the case of the rocket, these pieces correspond to the ejected propellent; if we were to measure the total mass of the rocket and its propellent, we would find that it is conserved. When the mass of a body is constant, Newton's second law becomes :

F = m \frac = m a

where a denotes the acceleration of the body. This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force. However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects A and B, with constant inertial masses m_A and m_B. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote F_AB, and the force exerted on B by A, which we denote F_BA. As we have seen, Newton's second law states that :

F_ = m_A a_A \,

and

F_ = m_B a_B \,

where a_A and a_B are the accelerations of A and B respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that :

F_ = - F_ \,

Substituting this into the previous equations, we obtain :

m_A = - \frac \, m_B

Note that our requirement that a_A be non-zero ensures that the fraction is well-defined. This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass m_B as (say) 1 kilogram. Then we can measure the mass of every other object in the universe by colliding it with the reference object and measuring the accelerations.

Gravitational mass

Gravitational mass is the mass of an object measured using the effect of a gravitational field on the object. The concept of gravitational mass rests on Newton's law of gravitation. Let us suppose we have two objects A and B, separated by a distance |r_AB|. The law of gravitation states that if A and B have gravitational masses M_A and M_B respectively, then each object exerts a gravitational force on the other, of magnitude :

|F| =

where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the acceleration of a reference mass at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is :

F = Mg \,

This is the basis by which masses are determined by weighing. In simple bathroom scales, for example, the force F is proportionate to the displacement of the spring beneath the weighing pan (see Hooke's law), and the scales are calibrated to take g into account, allowing the mass M to be read off.

Equivalence of inertial and gravitational masses

The equivalence of inertial and gravitational masses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have an object with inertial and gravitational masses m and M respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration :

K = \frac g

This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the universality of free-fall. (In addition, the constant K can be taken to be 1 by defining our units appropriately.) The first experiments demonstrating the universality of free-fall were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is unlikely to be true; actually, he performed his experiments with balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Roland Eötvös, using the torsion balance pendulum, in 1889. To date, no deviation from universality, and thus from Galilean equivalence, has ever been found. More precise experimental efforts are still being carried out. It should be noted that the universality of free fall only applies to systems in which gravity is the only force acting. All other forces, especially friction and air resistance, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height, we all know that the feather will take much longer to reach the ground. This happens because the feather is not really in free fall: the force of air resistance on it is about as strong as the force of gravity. On the other hand, if the experiment is performed in a vacuum, where there is no air resistance, the hammer and the feather should fall at the same rate and reach the ground together. This demonstration was, in fact, carried out in 1971 during the Apollo 15 Moon walk, by Commander David Scott. A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that inertial and gravitational masses are fundamentally the same thing. All of the predictions of general relativity, such as the curvature of spacetime, are ultimately derived from this principle.

Relativistic relation among mass, energy and momentum

Special relativity is a necessary extension of classical physics. In particular, special relativity succeeds where classical mechanics fails badly in describing objects moving at speeds close to the speed of light. In relativistic mechanics, the mass (m) of a free particle is related to its energy (E) and momentum (p) by the equation :

\frac = m^2 c^2 + p^2

. where c is the speed of light. This is sometimes referred to as the mass-energy-momentum relation. The first thing to notice about this equation is that it can cope with massless objects (m = 0), for which it reduces to :

E = pc \,

In classical mechanics, massless objects are an ill-defined concept, since applying any force to one would produce, via Newton's second law, an infinite acceleration - a nonsensical result. In relativistic mechanics, they are objects that are always traveling at the speed of light; an example being light itself, in the form of photons. The above equation says that the energy carried by a massless object is directly proportional to its momentum. Let us now consider objects with non-zero mass. For these, the quantity m has a simple physical meaning: it is the inertial mass of the object as measured in its rest frame, the frame of reference in which its velocity is zero. (Note: massless objects do not possess a rest frame; they are moving at the speed of light in any frame of reference.) The way we would measure m is exactly the same as in classical mechanics, which we described above: bouncing it off a reference object and measuring the accelerations. As long as the velocity of each object remains much smaller than the speed of light during this procedure, relativistic corrections to classical mechanics will be utterly negligible. In the rest frame, the velocity is zero, and thus so is the momentum p. The mass-energy-momentum relation thus reduces to :

E = mc^2 \,

which states that the energy of an object as measured in its rest frame - its "rest energy" - is equal to its mass times the square of the speed of light. Some books follow this up by stating that "mass and energy are equivalent", but this is somewhat misleading. The mass of an object, as we have defined it, is a quantity intrinsic to the object, and independent of our current frame of reference. The energy E, on the other hand, varies with the frame of reference; if the frame is moving at a high velocity relative to the object, E will be very large, simply because the object has a lot of kinetic energy in that frame. Thus, E = mc² is not a "good" relativistic statement; it is true only in the rest frame of the object. Some authors define a quantity known as the relativistic mass, which is basically the quantity E/c². This makes the "equivalence" of "mass" and energy true by definition, though neither quantity is frame-independent! "Relativistic mass" was used in many early writings on relativity, and it is still used in books for laymen as well as introductory physics classes. However, the concept is downplayed or discouraged by many physicists nowadays, for reasons explained in the article on relativistic mass. Following the modern usage, whenever we refer to "mass" in this article we always mean the rest mass, unless otherwise identified. Having defined the mass of an object, let us look at how it behaves when not at rest. We can arrange the mass-energy-momentum relation in the following way: :

E = mc^2 \sqrt

When the momentum p is much smaller than mc, we can Taylor expand the square root, with the result :

E = mc^2 +  + \cdots

The leading term, which is the largest, is of course the rest energy. The object always has this minimum amount of energy, regardless of its momentum. The second term is the classical expression for the kinetic energy of the particle, and the higher-order terms are basically relativistic corrections for the kinetic energy. Under normal circumstances, the rest energy of an object is inaccessible, in the sense that it cannot be used to do mechanical work. When the object hits something, it can do work by transferring its momentum, and thus its kinetic energy, to whatever it hit. However, the rest energy depends only on the mass of the object, which does not change during collisions, so it cannot be transferred along with the kinetic energy. On the other hand, it is possible to access the rest energy using processes that split or combine particles. The reason is that mass, as we have defined it, is not conserved during such processes. The simplest example is the process of electron-positron annihilation, in which an electron and a positron annihilate each other to produce a pair of photons: the electron and positron both have non-zero mass, but the photons are massless. Other examples include nuclear fusion and nuclear fission. Metabolism, fire and other exothermic chemical processes also convert mass to energy, however the mass change from these is negligible. Energy, unlike mass, is always conserved in special relativity, so, roughly speaking, what is happening in these reactions is that the rest energy of the reactants is being transformed into the kinetic energy of the reaction products. The fact that rest energy can be liberated in this way is one of the most important predictions of special relativity.

References

- R.V. Eötvös et al, Ann. Phys. (Leipzig) 68 11 (1922)

External links

- [http://math.ucr.edu/home/baez/physics Usenet Physics FAQ]
- [http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html Does mass change with velocity?]
- [http://math.ucr.edu/home/baez/physics/Relativity/SR/light_mass.html Does light have mass?]
- [http://www.teleles.nl/pdf/total_artikel.pdf Mass & energy]
- [http://www.geocities.com/physic1525/inertiaenergy.html The law of the inertia of the energy and the speed of the gravity. See chapter 3 The energy has mass ]
- [http://www.geocities.com/physics_world/stp/title.htm Dialog: Use and abuse of the concept of mass (from Spacetime physics by Edwin F. Taylor and John A. Wheeler)]
- [http://arxiv.org/PS_cache/physics/pdf/0111/0111134.pdf Photons, Clocks, Gravity and the Concept of Mass by L.B.Okun]
- [http://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html The Apollo 15 Hammer-Feather Drop]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units] Category:Physical chemistry Category:Classical mechanics
- ko:질량 ms:Jisim ja:質量 simple:Mass th:มวล

Torque

: For the game engine, see Torque Game Engine. For the item of jewelry, see torc. In physics, torque can be thought of informally as "rotational force". Torque is measured in units of newton metres. The concept of Torque, also called moment or couple, originated with the work of Archimedes on levers. The rotational analogues of force, mass and acceleration are torque, moment of inertia and angular acceleration respectively. The force applied to a lever, multiplied by its distance from the lever's fulcrum, is the torque. For example, a force of three newtons applied two metres from the fulcrum exerts the same torque as one newton applied six metres from the fulcrum. This assumes the force is in a direction at right angles to the straight lever. More generally, one may define torque as the cross product: :

\boldsymbol = \mathbf \times \mathbf

where r is the vector from the axis of rotation to the point on which the force is acting F is the vector of force.

Units

Torque has dimensions of distance × force and the SI units of torque are stated as "newton-metres". Even though the order of "newton" and "metre" are mathematically interchangeable, the BIPM (Bureau International des Poids et Mesures) specifies that the order should be N·m not m·N[http://www1.bipm.org/en/si/derived_units/2-2-2.html]. The joule, the SI unit for energy or work, is also defined as 1 N·m, but this unit is not used for torque. Of course, the dimensional equivalence of these units is not simply a coincidence; a torque of 1 N·m applied through a full revolution will require an energy of exactly 2π joules. Mathematically, :

E=\tau\theta

where E is the energy τ is torque θ is the angle moved, in radians. Other non-SI units of torque include "pound-force-feet" or "foot-pounds-force" or "ounce-force-inches or meter-kilograms-force.

Special cases and other facts

Moment arm formula

kilograms-force A very useful special case, often given as the definition of torque in fields other than physics, is as follows: :

\boldsymbol = (\textrm) \times \textrm

The construction of the "moment arm" is shown in the figure below, along with the vectors r and F mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in three-dimensional cases. If the force is perpendicular to the displacement vector r, the moment arm will be equal to the distance to the centre, and torque will be a maximum for the given force. The equation for the magnitude of a torque arising from a perpendicular force: :

\boldsymbol = (\textrm) \times \textrm

For example, if a person places a force of 10 N on a spanner which is 0.5m long, the torque will be 5 N·m, assuming that the person pulls the spanner in the direction best suited to turning bolts.

Force at an angle

If a force of magnitude F is at an angle θ from the displacement arm of length r (and within the plane perpendicular to the rotation axis), then from the definition of cross product, the magnitude of the torque arising is: :

\boldsymbol \tau=rF \sin \theta

Static equilibrium

For an object to be at static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques (moments) about any point. For a two-dimensional situation with horizontal and vertical forces, the sum of the forces requirement is two equations: ΣH = 0 and ΣV = 0, and the torque a third equation: Στ = 0. That is, to solve statically determinate equilibrium problems in two-dimensions, we use three equations.

Torque as a function of time

Torque is the time-derivative of angular momentum, just as force is the time derivative of linear momentum. For multiple torques acting simultaneously: :

\sum\boldsymbol =

where L is angular momentum. Angular momentum on a rigid body can be written in terms of its moment of inertia

\boldsymbol I

and its angular velocity

\boldsymbol

: :

\mathbf=I\,\boldsymbol

so if

\boldsymbol I

is constant, :

\boldsymbol=I=I\boldsymbol

where α is angular acceleration, a quantity usually measured in rad/s²

Machine torque

Torque is part of the basic specification of an engine: the power output of an engine is expressed as its torque multiplied by its rotational speed. Internal-combustion engines produce useful torque only over a limited range of rotational speeds (typically from around 1,000–6,000 rpm for a small car). The varying torque output over that range can be measured with a dynamometer, and shown as a torque curve. The peak of that torque curve usually occurs somewhat below the overall power peak. The torque peak cannot, by definition, appear at higher rpm than the power peak. Understanding the relationship between torque, power and engine speed is vital in automotive engineering, concerned as it is with transmitting power from the engine through the drive train to the wheels. The gearing of the drive train must be chosen appropriately to make the most of the motor's torque characteristics. Steam engines and electric motors tend to produce maximum torque at or around zero rpm, with the torque diminishing as rotational speed rises (due to increasing friction and other constraints). Therefore, these types of engines usually have quite different types of drivetrains from internal combustion engines. Torque is also the easiest way to explain mechanical advantage in just about every simple machine.

Relationship between torque and power

If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through a rotational distance, it is doing work. Power is the work per unit time. However, time and rotational distance are related by the angular speed where each revolution results in the circumference of the circle being travelled by the force that is generating the torque. This means that torque that is causing the angular speed to increase is doing work and the generated power may be calculated as: :

\mbox=\mbox \times \mbox

Mathematically, the equation may be rearranged to compute torque for a given power output. However in practice there is no direct way to measure power whereas torque and angular speed can be measured directly. Consistent units must be used. For metric SI units power is watts, torque is newton-metres and angular speed is radians per second (not rpm and not even revolutions per second).

Conversion to other units

For different units of power, torque or angular speed, a conversion factor must be inserted into the equation. For example, if the angular speed is measured in revolutions instead of radians, a conversion factor of

2 \pi

must be added because there are

2 \pi

radians in a revolution: :

\mbox=\mbox \times 2 \pi \times \mbox

, where rotational speed is in revolutions per unit time Some people (e.g. American automotive engineers) use horsepower (imperial mechanical) for power, foot-pounds (lbf·ft) for torque and rpm's (revolutions per minute) for angular speed. This results in the formula changing to: :

\mbox = \frac

This conversion factor is approximate because the transcendental number π appears in it; a more precise value is 5252.113 122 032 55... It also changes with the definition of the horsepower, of course; for example, using the metric horsepower, it becomes ~5180. Use of other units (e.g. BTU/h for power) would require a different custom conversion factor.

Derivation

For a rotating object, the linear distance covered at the circumference in a radian of rotation is the product of the radius with the angular speed. That is: linear speed = radius x angular speed. By definition, linear distance=linear speed x time=radius x angular speed x time. By the definition of torque: torque=force x radius. We can rearrange this to determine force=torque/radius. These two values can be substituted into the definition of power: :

\mbox = \frac=\frac  = \mbox \times \mbox

The radius r and time t have dropped out of the equation. However angular speed must be in radians, by the assumed direct relationship between linear speed and angular speed at the beginning of the derivation. If the rotational speed is measured in revolutions per unit of time, the linear speed and distance are increased proportionately by

2 \pi

in the above derivation to give: :

\mbox=\mbox \times 2 \pi \times \mbox

If torque is in lbf·ft and rotational speed in revolutions per minute, the above equation gives power in ft·lbf/min. The horsepower form of the equation is then derived by applying the conversion factor 33,000 ft·lbf/min per horsepower: :

\mbox=\mbox \times 2 \pi \times \mbox \frac \times \frac = \frac

Because

5252.113... = 33,000/2 \pi

References

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- Category:Mechanical engineering Category:Physical quantity Category:Introductory physics Category:Rotational symmetry ms:Tork ja:力のモーメント

Motion

:This article is about motion in physics. See also motion (legal), motion (democracy) and Apple Motion. In physics, motion means a change in the position of a body with respect to time, as measured by a particular observer in a particular frame of reference. Until the end of the 19th century, Newton's laws of motion, which he posited as axioms or postulates in his famous Principia, were the basis of what has since become known as classical physics. Calculations of trajectories and forces of bodies in motion based on Newtonian or classical physics were very successful until physicists began to be able to measure and observe very fast physical phenomena. At very high speeds, the equations of classical physics were not able to calculate accurate values. To address these problems, the ideas of Henri Poincaré and Albert Einstein concerning the fundamental phenomenon of motion were adopted in lieu of Newton's. Whereas Newton's laws of motion assumed absolute values of space and time in the equations of motion, the model of Einstein and Poincaré, now called the special theory of relativity, assumed values for these concepts with arbitrary zero points. Because (for example) the special relativity equations yielded accurate results at high speeds and Newton's did not, the special relativity model is now accepted as explaining bodies in motion (when we ignore gravity). However, as a practical matter, Newton's equations are much easier to work with than those of special relativity and therefore are more often used in applied physics and engineering. In the newtonian model, because motion is defined as the proportion of space to time, these concepts are prior to motion, just as the concept of motion itself is prior to force. In other words, the properties of space and time determine the nature of motion and the properties of motion, in turn, determine the nature of force. In the special relativistic model, motion can be thought of as something like an angle between a space direction and the time direction. In special relativity and euclidean space, only relative motion can be measured and that absolute motion is meaningless.

Kinetic energy

This article is about kinetic energy. For Kinetic wristwatches, see Seiko Kinetic watches Kinetic energy is energy that a body has as a result of its speed or energy of motion. It is formally defined as work needed to accelerate a body from rest to a velocity v. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work would also be required to return the body to a state of rest from that velocity. =Simple explanation= Energy can exist in many forms, for example chemical energy, heat, electromagnetic radiation, potential energy (both gravitational and elastic), nuclear energy and kinetic energy. These forms of energy can often be converted to other forms. Kinetic energy can be best understood by examples that demonstrate how it is transformed from other forms of energy and to the other forms. For example a cyclist will use chemical energy that was provided by food to accelerate a bicycle to a chosen velocity. This velocity can be maintained without further work, except to overcome air-resistance and friction. The energy has been converted into the energy of motion, known as kinetic energy but the process is not completely efficient and heat is also produced within the cyclist. The kinetic energy in the moving bicycle and the cyclist can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. (There are some frictional losses so that the bicycle will never quite regain all the original speed.) Alternatively the cyclist could connect a dynamo to one of the wheels and also generate some electrical energy on the descent. The bicycle would be travelling more slowly at the bottom of the hill because some of the energy has been diverted into making electrical power. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated as heat energy. See also energy conversion.

Simple calculation

The kinetic energy of a body =

\begin \frac \end mv^2

where m is the mass and v is the velocity of the body. Note that the kinetic energy increases with the square of the velocity. This means for example that if you are traveling twice as fast, you need to lose four times as much energy to stop. The braking distance of a car at 100 km/h is therefore four times as far as the braking distance at 50 km/h.

More simple examples

The space shuttle uses chemical energy to take off and gains considerable kinetic energy because it must reach orbital velocity. This kinetic energy gained during launch will remain constant while the shuttle is in orbit because there is almost no friction. However it becomes apparent at re-entry when the kinetic energy is converted to heat. Kinetic energy can be passed from one object to another. In the game of billiards, the player gives kinetic energy to the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it will slow down dramatically and the ball it collided with will accelerate to a velocity as the kinetic energy is passed on to it. Collisions in billiards are elastic collisions, where kinetic energy is preserved. Flywheels are being developed as a method of energy storage (see article flywheel energy storage). This illustrates that kinetic energy can also be rotational. Note the formula in the articles on flywheels for calculating rotational kinetic energy is different, though analogous. =Rigorous definitions= :

E_k = \int \mathbf \cdot \mathrm\mathbf = \int \mathbf \cdot \mathrm\mathbf  = (1/2)mv^2

the words in the above equation state that the kinetic energy (E_k) is equal to the integral of the dot product of the velocity (v) of a body and the infinitesimal of the body's momentum (p).

In Newtonian mechanics

For non-relativistic mechanics, the total kinetic energy of a body can be considered as the sum of the body's translational kinetic energy and its rotational energy, or angular kinetic energy: :

E_k = E_t + E_r \,

where: :E_k is the total kinetic energy :E_t is the translational kinetic energy :E_r is the rotational energy or angular kinetic energy For the translational kinetic energy of a body with constant mass m, whose centre of mass is moving in a straight line with linear velocity v, as seen above is equal to :

E_t = \begin \frac \end mv^2

where: :m is mass of the body :v is linear velocity of the centre of mass body Thus, for a speed of 10 m/s the kinetic energy is 50 J/kg, for a speed of 10 km/s it is 50 MJ/kg, etc. If a body is rotating, its rotational kinetic energy or angular kinetic energy is simply sum of kinetic energies of its moving parts, and thus is equal to: :

E_r = \begin \frac \end I \omega^2

where: :I is the body's moment of inertia :ω is the body's angular velocity. The kinetic energy of a system depends on the inertial frame of reference. It is lowest with respect to the center of mass, i.e., in a frame of reference in which the center of mass is stationary. In another frame of reference the additional kinetic energy is that corresponding to the total mass and the speed of the center of mass.

In Relativistic mechanics

In Einstein's relativistic mechanics, (used especially for near-light velocities) mass no longer stays constant and accurate calculation of work to accelerate body results in the following expression for kinetic energy: :

E_k = m c^2 (\gamma - 1) = \gamma m c^2 - m c^2 \;\!

\gamma = \frac

E_k = \gamma m c^2 - m c^2 
= \left( \frac - 1 \right) m c^2

where: :E_k is the kinetic energy of the body :v is the velocity of the body :m is its rest mass :c is the speed of light in a vacuum :γmc² is the total energy of the body :mc² is the rest mass energy (90 petajoule/kg) It is an edifying exercise to show that the ratio of this relativistic kinetic energy to the Newtonian kinetic energy given by (1/2)mv^{2 approaches 1 as v approaches 0, i.e.,

: $\lim_=1$

This can be done by the techniques of first-year calculus.

Relativity theory states that the kinetic energy of an object grows towards infinity as its velocity approaches the speed of light, and thus that it is impossible to accelerate an object to this boundary.

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.

The next term in the approximation is 0.375 mv⁴/c², e.g. for a speed of 10 km/s this is 0.04 J/kg, for a speed of 100 km/s it is 40 J/kg, etc.

The exact Taylor series is
: $E_k = mv^2 + m\left(\right) + m\left(\right) + \dots \,$

= See also =

- Recoil

- Joule

- Kinetic projectile

= References =

-

-

-

Category:Energy
Category:Ammunition
Category:Classical mechanics
Category:Introductory physics

ko:운동 에너지

ms:Tenaga kinetik

ja:運動エネルギー

simple:Kinetic energy

Flywheel

]
A flywheel is a heavy rotating disk used as a repository for angular momentum. Flywheels resist changes in their rotation speed, which helps steady the rotation of the shaft when an uneven torque is exerted on it by its power source such as a piston-based, (reciprocating) engine, or when the load placed on it is intermittent (such as a piston-based pump). Flywheels can also be used by small motors to store up energy over a long period of time and then release it over a shorter period of time, temporarily magnifying its power output for that brief period. Recently, flywheels have become the subject of extensive research as power storage devices; see flywheel energy storage.

A momentum wheel is a type of flywheel useful in satellite pointing operations, in which the flywheels are used to point the satellite's instruments in the correct directions without the use of thrusters.

The kinetic energy stored in a rotating flywheel is

: $E = \frac I \omega^2$

where $I$ is the moment of inertia of the mass about the center of rotation and $\omega$ (omega) is the angular velocity in radian units. A flywheel is more effective when its inertia is larger, as when its mass is located farther from the center of rotation either due to a more massive rim or due to a larger diameter. Note the similarity of the above formula to the kinetic energy formula E = mv²/2, where linear velocity v is comparable to the rotational velocity, and the mass is comparable to the rotational inertia.

The flywheel has been used since ancient times, the most common traditional example being the potter's wheel. In the Industrial Revolution, James Watt contributed to the development of the flywheel in the steam engine.

See also

- List of energy topics

- Gyroscope

- Momentum wheel

- Regenerative braking

- Flywheel energy storage

External links

- [http://peswiki.com/index.php/PowerPedia:Flywheels Flywheels] - index at PESWiki.com

Category:Energy storage
Category:Engine technology

Cross product
In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. It is also known as the vector product or outer product. It differs from the dot product in that it results in a pseudovector rather than in a scalar. Its main use lies in the fact that the cross product of two vectors is orthogonal to both of them.

Definition

The cross product of the two vectors a and b is denoted by a × b (in longhand some mathematicians write a∧b to avoid confusion with the letter x - this should not be confused with the logical and operator, $\and$ ). It can be defined by

: $\mathbf \times \mathbf = \mathbf\hat \left| \mathbf \right| \left| \mathbf \right| \sin \theta$

where θ is the measure of the angle between a and b (0° ≤ θ ≤ 180°) on the plane defined by the span of the vectors, and n is a unit vector perpendicular to both a and b.

The problem with this definition is that there are two unit vectors perpendicular to both a and b: if n is perpendicular, then so is −n.

Which vector is the "correct" one by convention depends upon the orientation of the vector space—i.e., on the handedness of the given orthogonal coordinate system (i, j, k). The cross product a × b is defined in such a way that (a, b, a × b) becomes right-handed if (i, j, k) is right-handed, or left-handed if (i, j, k) is left-handed.

An easy way to compute the direction of the resultant vector is the "right-hand rule." If the coordinate system is right-handed, one simply points the forefinger in the direction of the first operand and the middle finger in the direction of the second operand. Then, the resultant vector is coming out of the thumb.

Because the cross product depends on the choice of coordinate system, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the “handedness” of the coordinate system is undone by a second cross product.

The cross product can be represented graphically, with respect to a right-handed coordinate system, as shown in the picture below.

Image:crossproduct.png

Properties

Geometric meaning

The length of the cross product, can be interpreted as the area of the parallelogram having a and b as sides:

:: $|a \times b|$

This means that the magnitude of the triple product gives the volume V of the parallelepiped formed by a, b, and c:

: $V = |\mathbf\cdot(\mathbf \times \mathbf)|.$

Algebraic properties

The cross product is anticommutative,
:a × b = -b × a,

distributive over addition,
:a × (b + c) = a × b + a × c,

and compatible with scalar multiplication so that
:(ra) × b = a × (rb) = r(a × b).

It is not associative, but satisfies the Jacobi identity:
:a × (b × c) + b × (c × a) + c × (a × b) = 0.

The distributivity, linearity and Jacobi identity show that R³ together with vector addition and cross product forms a Lie algebra.

Further, two non-zero vectors a and b are parallel iff a × b = 0.

Associativity

The vector cross product is an example of a non-associative map. In general
: $\left(\mathbf\times\mathbf\right)\times\mathbf\neq
\mathbf\times\left(\mathbf\times\mathbf\right).$

To see this, consider the case where $\mathbf$ and $\mathbf$ are parallel to one another. Then the left hand side is zero, and the right hand side is (in general) non-zero.

Matrix notation

The unit vectors i, j, and k from the given orthogonal coordinate system satisfy the following equalities:

:i × j = k j × k = i k × i = j.

With these rules, the coordinates of the cross product of two vectors can be computed easily, without the need to determine any angles: Let

:a = a₁i + a₂j + a₃k = [a₁, a₂, a₃]

and

:b = b₁i + b₂j + b₃k = [b₁, b₂, b₃].

Then
:a × b = [a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁].

The above component notation can also be written formally as the determinant of a matrix:

: $\mathbf\times\mathbf=\det \begin
\mathbf & \mathbf & \mathbf \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end.$

The determinant of three vectors can be recovered as
:det (a, b, c) = a · (b × c).

Intuitively, the cross product can be described by Sarrus's scheme. Consider the table
: $\begin
\mathbf & \mathbf & \mathbf & \mathbf & \mathbf & \mathbf \\
a_1 & a_2 & a_3 & a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 & b_1 & b_2 & b_3
\end$
For the first three unit vectors, multiply the elements on the diagonal to the right (e.g. the first diagonal would contain i, a₂, and b₃). For the last three unit vectors, multiply the elements on the diagonal to the left and then negate the product (e.g. the last diagonal would contain k, a₂, and b₁). The cross product would be defined by the sum of these products:
: $\mathbf(a_2b_3) + \mathbf(a_3b_1) + \mathbf(a_1b_2) - \mathbf(a_3b_2) - \mathbf(a_1b_3) - \mathbf(a_2b_1).$

Although written here in terms of coordinates, it follows from the geometrical definition above that the cross product is invariant under rotations about the axis defined by $\mathbf\times\mathbf$ , and inverses under swapping $\mathbf$ and $\mathbf.$

The cross product can also be described in terms of quaternions. Notice for instance that the above given cross product relations among i, j, and k agree with the multiplicative relations among the quaternions i, j, and k. In general, if we represent a vector [a₁, a₂, a₃] as the quaternion a₁i + a₂j + a₃k, we obtain the cross product of two vectors by taking their product as quaternions and deleting the real part of the result (the real part will be the negative of the dot product of the two vectors). More about the connection between quaternion multiplication, vector operations and geometry can be found at quaternions and spatial rotation.

Lagrange's formula

While this is not strictly a property of the cross-product, it is an identity involving the cross-product which is very useful. It is written as

:a × (b × c) = b(a · c) − c(a · b),

which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. A special case, regarding gradients and useful in vector calculus, is given below.
: $\begin
\nabla \times (\nabla \times \mathbf)
&=& \nabla (\nabla \cdot \mathbf )
- (\nabla \cdot \nabla) \mathbf \\
&=& \mbox(\mbox \mathbf )
- \mbox \mathbf.
\end$
This is a special case of the more general Laplace-de Rham operator $\Delta = d \delta + \delta d$ .

Another useful identity of Lagrange is
: $|a \times b|^2 + |a \cdot b|^2 = |a|^2 |b|^2.$
This is a special case of the multiplicativity $|vw| = |v| |w|$ of the norm in the quaternion algebra.

Applications

The cross product occurs in the formula for the vector operator curl.
It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product.

The cross product can also be used to calculate the normal for a triangle or polygon.

Given a point p and a line through a and b in a plane, all with z coordinate zero, then the z component of (p-a) × (b-a) will be positive or negative, depending on which side of the line p is.

Higher dimensions

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. See seven dimensional cross product for the main article.

In general dimension, there is no direct analogue of the binary cross product. There is however the wedge product, which
has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector. The cross product can be interpreted as the wedge product in three dimensions after using Hodge duality to identify 2-vectors with vectors.

One can also construct an n-ary analogue of the cross product in Rⁿ⁺¹ given by
: $\bigwedge(\mathbf_1,\cdots,\mathbf_n)=
\begin
v_1^1 &\cdots &v_1^\\
\vdots &\ddots &\vdots\\
v_n^1 & \cdots &v_n^\\
\mathbf_1 &\cdots &\mathbf_
\end.$

This formula is identical in structure to the determinant formula for the normal cross product in R³ except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v₁,...,v_n,Λ(v₁,...,v_n)) have a positive orientation with respect to (e₁,...,e_n+1). If n is even, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is odd, however, the distinction must be kept. This n-ary form enjoys many of the same properties as the vector cross product: it is alternating and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments.

The wedge product and dot product can be combined to form the Clifford product.

In the context of multilinear algebra, it is possible to define a generalized cross product in terms of parity such that the generalized cross product between two vectors of dimension n is a tensor of rank n−2. This is a different concept than what is discussed above.

Symbol
The cross-product symbol, ×, is × in HTML and U+00D7 in Unicode.

See also

- Right-handed rule

- Evaluating cross products

- Lagrange's formula - a common useful cross product identity.

- Triple product - a formula using the cross product and dot product

- Dot product

External links

- [http://www.hkshum.net/Math Vector Cross Product] which allows you to cross two 3D vectors. Look under the Vector Cross Product heading.

- [http://uk.arxiv.org/abs/math.la/0204357 Multi-dimensional vector product] is only possible in seven dimensional space.
Category:Abstract algebra
Category:Linear algebra

ko:외적

ja:クロス積

Mass
:For other senses of this word, see mass (disambiguation).

Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. Unlike weight, the mass of something stays the same regardless of location. It is a central concept of classical mechanics and related subjects.

Strictly speaking, there are three different quantities called mass:

- Inertial mass is a measure of an object's inertia: 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.

- Passive gravitational mass is a measure of the strength of an object's interaction with the gravitational field. Within the same gravitational field, an object with a smaller passive gravitational mass experiences a smaller force than an object with a larger passive gravitational mass. (This force is called the weight of the object. In informal usage, the word "weight" is often used synonymously with "mass", because the strength of the gravitational field is roughly constant everywhere on the surface of the Earth. In physics, the two terms are distinct: an object will have a larger weight if it is placed in a stronger gravitational field, but its passive gravitational mass remains unchanged.)

- Active gravitational mass is a measure of the strength of the gravitational field due to a particular object. For example, the gravitational field that one experiences on the Moon is weaker than that of the Earth because the Moon has less active gravitational mass.

Introduction

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. One of the consequences of the equivalence of inertial mass and passive gravitational mass is the fact, famously demonstrated by Galileo Galilei, that objects with different masses fall at the same rate, assuming factors like air resistance are negligible. The theory of general relativity, the most accurate theory of gravitation known to physicists to date, rests on the assumption that inertial and passive gravitational mass are completely equivalent. This is known as the weak equivalence principle. Classically, active and passive gravitational mass were equivalent as a consequence of Newton's third law, but a new axiom is required in the context of relativity's reformulation of gravity and mechanics. Thus, standard general relativity also assumes the equivalence of inertial mass and active gravitational mass; this equivalence is sometimes called the strong equivalence principle.

If one were to treat inertial mass m_i, passive gravitational mass m_p, and active gravitational mass m_a distinctly, Newton's law of universal gravitation would give as force on the second mass due to the first mass
: $m_a_2=\frac.$

Newton's third law, of reciprocal actions, shows that active and passive mass are proportional. As a result they can be defined to be equal.

Units of mass

In the SI system of units, mass is measured in kilograms (kg). Many other units of mass are also employed, such as: grams (g), metric tons, pounds, ounces, long and short tons, quintals, slugs, atomic mass units, Planck masses, solar masses, and eV/c².

The eV/c² unit is based on the electron volt (eV), which is normally used as a unit of energy. However, because of the relativistic connection between (rest) mass and energy, E = mc² (see below), it is possible to use any unit of energy as a unit of mass instead. Thus, in particle physics where mass and energy are often interchanged, it is common to use not only eV/c² but even simply eV as a unit of mass (roughly 1.783 × 10^-36 kg).

Because the gravitational acceleration is approximately constant on the surface of the Earth, a unit like the pound is often used to measure either mass or force (e.g. weight), although the pound is officially defined as a unit of mass.

For more information on the different units of mass, see Orders of magnitude (mass).

Inertial mass

Inertial mass is the mass of an object measured by its resistance to acceleration.

To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "mass" in any essential way.

According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion

: $F = \frac (mv)$

where F is the force acting on the body and v is its velocity. For the moment, we will put aside the question of what "force acting on the body" actually means.

Now, suppose that the mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. These are very reasonable assumptions for everyday objects, though, as we will see, the situation gets more complicated when we take special relativity into account. Another point to note is that, even in classical mechanics, it is sometimes useful to treat the mass of an object as changing with time. For example, the mass of a rocket decreases as the rocket fires. However, this is an approximation, based on ignoring pieces of matter which enter or leave the system. In the case of the rocket, these pieces correspond to the ejected propellent; if we were to measure the total mass of the rocket and its propellent, we would find that it is conserved.

When the mass of a body is constant, Newton's second law becomes

: $F = m \frac = m a$

where a denotes the acceleration of the body.

This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects A and B, with constant inertial masses m_A and m_B. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote F_AB, and the force exerted on B by A, which we denote F_BA. As we have seen, Newton's second law states that

: $F_ = m_A a_A \,$ and $F_ = m_B a_B \,$

where a_A and a_B are the accelerations of A and B respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that

: $F_ = - F_ \,$

Substituting this into the previous equations, we obtain

: $m_A = - \frac \, m_B$

Note that our requirement that a_A be non-zero ensures that the fraction is well-defined.

This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass m_B as (say) 1 kilogram. Then we can measure the mass of every other object in the universe by colliding it with the reference object and measuring the accelerations.

Gravitational mass

Gravitational mass is the mass of an object measured using the effect of a gravitational field on the object.

The concept of gravitational mass rests on Newton's law of gravitation. Let us suppose we have two objects A and B, separated by a distance |r_AB|. The law of gravitation states that if A and B have gravitational masses M_A and M_B respectively, then each object exerts a gravitational force on the other, of magnitude

: $|F| =$

where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the acceleration of a reference mass at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

: $F = Mg \,$

This is the basis by which masses are determined by weighing. In simple bathroom scales, for example, the force F is proportionate to the displacement of the spring beneath the weighing pan (see Hooke's law), and the scales are calibrated to take g into account, allowing the mass M to be read off.

Equivalence of inertial and gravitational masses

The equivalence of inertial and gravitational masses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have an object with inertial and gravitational masses m and M respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration

: $K = \frac g$

This says that the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the universality of free-fall. (In addition, the constant K can be taken to be 1 by defining our units appropriately.)

The first experiments demonstrating the universality of free-fall were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is unlikely to be true; actually, he performed his experiments with balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Roland Eötvös, using the torsion balance pendulum, in 1889. To date, no deviation from universality, and thus from Galilean equivalence, has ever been found. More precise experimental efforts are still being carried out.

It should be noted that the universality of free fall only applies to systems in which gravity is the only force acting. All other forces, especially friction and air resistance, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height, we all know that the feather will take much longer to reach the ground. This happens because the feather is not really in free fall: the force of air resistance on it is about as strong as the force of gravity. On the other hand, if the experiment is performed in a vacuum, where there is no air resistance, the hammer and the feather should fall at the same rate and reach the ground together. This demonstration was, in fact, carried out in 1971 during the Apollo 15 Moon walk, by Commander David Scott.

A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that inertial and gravitational masses are fundamentally the same thing. All of the predictions of general relativity, such as the curvature of spacetime, are ultimately derived from this principle.

Relativistic relation among mass, energy and momentum

Special relativity is a necessary extension of classical physics. In particular, special relativity succeeds where classical mechanics fails badly in describing objects moving at speeds close to the speed of light.

In relativistic mechanics, the mass (m) of a free particle is related to its energy (E) and momentum (p) by the equation

: $\frac = m^2 c^2 + p^2$ .

where c is the speed of light. This is sometimes referred to as the mass-energy-momentum relation.

The first thing to notice about this equation is that it can cope with massless objects (m = 0), for which it reduces to

: $E = pc \,$

In classical mechanics, massless objects are an ill-defined concept, since applying any force to one would produce, via Newton's second law, an infinite acceleration - a nonsensical result. In relativistic mechanics, they are objects that are always traveling at the speed of light; an example being light itself, in the form of photons. The above equation says that the energy carried by a massless object is directly proportional to its momentum.

Let us now consider objects with non-zero mass. For these, the quantity m has a simple physical meaning: it is the inertial mass of the object as measured in its rest frame, the frame of reference in which its velocity is zero. (Note: massless objects do not possess a rest frame; they are moving at the speed of light in any frame of reference.) The way we would measure m is exactly the same as in classical mechanics, which we described above: bouncing it off a reference object and measuring the accelerations. As long as the velocity of each object remains much smaller than the speed of light during this procedure, relativistic corrections to classical mechanics will be utterly negligible.

In the rest frame, the velocity is zero, and thus so is the momentum p. The mass-energy-momentum relation thus reduces to

: $E = mc^2 \,$

which states that the energy of an object as measured in its rest frame - its "rest energy" - is equal to its mass times the square of the speed of light.

Some books follow this up by stating that "mass and energy are equivalent", but this is somewhat misleading. The mass of an object, as we have defined it, is a quantity intrinsic to the object, and independent of our current frame of reference. The energy E, on the other hand, varies with the frame of reference; if the frame is moving at a high velocity relative to the object, E will be very large, simply because the object has a lot of kinetic energy in that frame. Thus, E = mc² is not a "good" relativistic statement; it is true only in the rest frame of the object.

Some authors define a quantity known as the relativistic mass, which is basically the quantity E/c². This makes the "equivalence" of "mass" and energy true by definition, though neither quantity is frame-independent! "Relativistic mass" was used in many early writings on relativity, and it is still used in books for laymen as well as introductory physics classes. However, the concept is downplayed or discouraged by many physicists nowadays, for reasons explained in the article on relativistic mass. Following the modern usage, whenever we refer to "mass" in this article we always mean the rest mass, unless otherwise identified.

Having defined the mass of an object, let us look at how it behaves when not at rest. We can arrange the mass-energy-momentum relation in the following way:

: $E = mc^2 \sqrt$

When the momentum p is much smaller than mc, we can Taylor expand the square root, with the result

: $E = mc^2 + + \cdots$

The leading term, which is the largest, is of course the rest energy. The object always has this minimum amount of energy, regardless of its momentum. The second term is the classical expression for the kinetic energy of the particle, and the higher-order terms are basically relativistic corrections for the kinetic energy.

Under normal circumstances, the rest energy of an object is inaccessible, in the sense that it cannot be used to do mechanical work. When the object hits something, it can do work by transferring its momentum, and thus its kinetic energy, to whatever it hit. However, the rest energy depends only on the mass of the object, which does not change during collisions, so it cannot be transferred along with the kinetic energy.

On the other hand, it is possible to access the rest energy using processes that split or combine particles. The reason is that mass, as we have defined it, is not conserved during such processes. The simplest example is the process of electron-positron annihilation, in which an electron and a positron annihilate each other to produce a pair of photons: the electron and positron both have non-zero mass, but the photons are massless. Other examples include nuclear fusion and nuclear fission. Metabolism, fire and other exothermic chemical processes also convert mass to energy, however the mass change from these is negligible.

Energy, unlike mass, is always conserved in special relativity, so, roughly speaking, what is happening in these reactions is that the rest energy of the reactants is being transformed into the kinetic energy of the reaction products. The fact that rest energy can be liberated in this way is one of the most important predictions of special relativity.

References

- R.V. Eötvös et al, Ann. Phys. (Leipzig) 68 11 (1922)

See also

- Density

- Higgs boson

- Orders of magnitude (mass)

- Planck units

- Volume

- Weight

External links

- [http://math.ucr.edu/home/baez/physics Usenet Physics FAQ]

- [http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html Does mass change with velocity?]

- [http://math.ucr.edu/home/baez/physics/Relativity/SR/light_mass.html Does light have mass?]

- [http://www.teleles.nl/pdf/total_artikel.pdf Mass & energy]

- [http://www.geocities.com/physic1525/inertiaenergy.html The law of the inertia of the energy and the speed of the gravity. See chapter 3 The energy has mass ]

- [http://www.geocities.com/physics_world/stp/title.htm Dialog: Use and abuse of the concept of mass (from Spacetime physics by Edwin F. Taylor and John A. Wheeler)]

- [http://arxiv.org/PS_cache/physics/pdf/0111/0111134.pdf Photons, Clocks, Gravity and the Concept of Mass by L.B.Okun]

- [http://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html The Apollo 15 Hammer-Feather Drop]

- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]

Category:Physical chemistry
Category:Classical mechanics

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ko:질량

ms:Jisim

ja:質量

simple:Mass

th:มวล

Angular velocity

The angular velocity of a point particle or rigid body describes the rate at which its orientation changes. It is analogous to translational velocity, and is defined in terms of the derivative of orientation with respect to time, just as translational velocity is the derivative of displacement with respect to time. It is customary to introduce the concept of velocity by first defining average velocity as displacement divided by time. There the analogy with angular velocity is less useful: for example, if a body is rotating at a constant angular velocity of one revolution per minute, then over a one-minute period the 'average angular velocity' of the body is zero, because the orientation is exactly the same at the beginning of the time period as it is at the end.

More precisely, if $A(t)$ is the special orthogonal linear transformation which describes the orientation, the angular velocity is defined as $A(t)^A(t)$ . It follows that angular velocity is a skew-adjoint linear transformation . It is useful to restrict attention to two or three dimensions and represent the three-dimensional Lie algebra of skew-adjoint linear transformations of V $_3$ (R) by R³. The commutator operation, which is the Lie product of the algebra, is represented by the cross product in R³. The rest of this article is devoted to a discussion in that style.

Vector angular velocity.
Angular velocity is the vector physical quantity that represents the process of rotation (change of orientation) that occurs at an instant of time. For a rigid body it supplements translational velocity of the center of mass to describe the full motion. It is usually represented by the symbol omega (Ω or ω). The magnitude of the angular velocity is the angular speed (or angular frequency) and is denoted by ω. The line of direction of the angular velocity is given by the axis of rotation, and the right hand rule indicates the positive direction, namely:

:If you curl the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb.

In SI units, angular velocity is measured in radians per second, (rad/s), although a direction must also be given. The dimensions of angular velocity are T ^-1, since radians are dimensionless.

For any particle of a moving and spinning body we have

: $\mathbf = \mathbf_t + \boldsymbol\omega \times (\mathbf - \mathbf_c)$

where $\mathbf$ is the total velocity of the particle, $\mathbf_t$ the translational velocity, $\mathbf$ the position of the particle, and $\mathbf_c$ the position of the center of the body.

To describe the motion the "center" can be any particle of the body or imaginary point that is rigidly connected to the body (the translation vector depends on the choice) but typically the center of mass is chosen, because it simplifies some formulas.

When the cross product is written in matrix form we have a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements; see also above.

With constant angular acceleration, the angular velocity conforms to the rotational equations of motion, equivalent to the standard linear equations of motion under constant linear acceleration.

Angular frequency is also used instead of normal frequency in some situations that don't actually involve rotation especially in electronics as it makes the expression of sinusoids and various equations that are obtained by calculus on sinusoids simpler. (ωt rather than 2πft).

The non-circular motion case
If the motion of a particle is described by a position vector-valued function r(t) — with respect to a fixed origin — then the angular velocity vector is
: $\boldsymbol\omega = \qquad \qquad (1)$
where
: $\mathbf(t) = \mathbf(t)$
is the linear velocity vector. Equation (1) is applicable to non-circular motions, e.g. elliptic orbits.

Derivation
Vector v can be resolved into a pair of components: $\mathbf_\perp$ which is perpendicular to r, and $\mathbf_\|$ which is parallel to r. The motion of the parallel component is completely linear and produces no rotation of the particle (with regard to the origin), so for purposes of finding the angular velocity it can be ignored. The motion of the perpendicular component is completely circular, since it is perpendicular to the radial vector, just like any tangent to a point on a circle.

The perpendicular component has magnitude
: $|\mathbf_\perp| = \qquad \qquad (2)$
where the vector $\mathbf \times \mathbf$ represents the area of the parallelogram two of whose sides are the vectors r and v. Dividing this area by the magnitude of r yields the height of this parallelogram between r and the side of the parallelogram parallel to r. This height is equal to the component of v which is perpendicular to r.

In the case of pure circular motion, the angular velocity is equal to linear velocity divided by the radius. In the case of generalized motion, the linear velocity is replaced by its component perpendicular to r, viz.
: $\omega = \qquad \qquad (3)$
therefore, putting equations (2) and (3) together yields
: $\omega = = |\boldsymbol\omega|. \qquad \qquad (4)$

Equation (4) gives the magnitude of the angular velocity vector. The vector's direction is given by its normalized version:
: $\hat\boldsymbol\omega = . \qquad \qquad (5)$
Then the entire angular velocity vector is given by putting together its magnitude and its direction:
: $\boldsymbol\omega = \omega \hat\boldsymbol\omega$
which, due to equations (4) and (5), is equal to
: $\boldsymbol\omega = ,$
which was to be demonstrated.

See also

- (Introductory)

- Angular momentum

- Angular frequency

- Areal velocity

- (Advanced)

- Isometry

- Orthogonal group

- Rotation group

- Lie algebra

External link

- [http://math.ucr.edu/home/baez/classical/galilei2.pdf Rotations and Angular Momentum] on the Classical Mechanics page of [http://math.ucr.edu/home/baez/README.html the website of John Baez], especially Questions 1 and 2.

Reference

- Peter M. Neumann; Gabrielle A. Stoy; Edward C. Thompson. Groups and Geometry, Oxford 1994, ISBN 01798534515. See pp. 108-110, 163-165 .

Category:Physical quantity
Category:Rotational symmetry
Category:Angle

ja:角速度
ms:Halaju angular

Moment of inertia
Moment of inertia quantifies the rotational inertia of an object, i.e. its inertia with respect to rotational motion, in a manner somewhat analogous to how mass quantifies the inertia of an object with respect to translational motion.

In general, an object's moment of inertia depends on its shape and the distribution of mass within that shape: the greater the concentration of material away from the object's centroid, the larger the moment of inertia. It also varies depending upon the axis of rotation specified; values relative to the object's centroid are typically taken as baseline values. See the list of moments of inertia for specific examples. The parallel axes theorem can be used to determine moments of inertia relative to displaced axes of rotation.

Rotational versions of Newton's second law and the formulas for momentum and kinetic energy, use moment of inertia in place of the mass of an object (with torque, angular velocity and angular acceleration replacing force, velocity and acceleration, respectively).

Moment of inertia is often represented by the letter I (capital i).

It should not be confused with the second moment of area or area moment of inertia, which is a property of a shape that is used to predict its resistance to bending and deflection. Most structural engineers, however, do refer to the second moment of area or area moment of inertia as simply the moment of inertia.
Unit
The SI unit for moment of inertia is kilogram metre squared (kg m²)

Derivation for point mass

Say we wanted to develop a formula for rotational kinetic energy that is analogous to the linear kinetic energy formula for a point mass.

An object with mass $m$ that is moving with velocity $v$ has a kinetic energy (KE) given as:

: $KE = \frac m v^2$

If an object is rotating at an angular velocity $\omega$ in a circle of radius $r$ , its linear velocity, v is equal to $\omega r$ . Substituting $v = \omega r$ into the kinetic energy equation above, we have:

: $KE = \frac m r^2 \omega^2$

Drawing parallels to the linear kinetic energy equation, we see that, for a rotating point mass, we substitute angular velocity, $\omega$ for linear velocity, $v$ . That leaves us with the quantity $m r^2$ , which takes the place of the object's mass in the linear kinetic energy equation. We give this quantity, a kind of "rotational mass", a name: moment of inertia.

Mathematical definition

For a small (pointlike) mass m located at distance r from axis of rotation the moment of inertia (versus this axis) is defined by:

: $I = mr^2\,$

For system of small masses moment of inertia is defined as the sum of moments of all parts:
: $I = \sum_i m_i r_i^2$

Continuous mass distributions require an infinite sum over all the point mass moments which make up the whole. This is accomplished by integrating all the masses $dm \,\!$ over all three-dimensional space involved:
: $I = \int r^2\,dm \,\!$

$dm \,\!$ is defined by the spatial density distribution $\rho \,\!$ .
: $dm=\rho\,dV \,\!$

Types of moment of inertia

There are an infinite number of moments of inertia for any object, one for every possible axis of rotation through the object's centroid. For convenience, the three moments of inertia typically used for an object are about axes parallel to the three Cartesian axes (X, Y, and Z):

: $I = \;$ moment of inertia about the current axis of rotation

: $I_ = \;$ moment of inertia about the line through the centroid, parallel to the X-axis

: $I_ = \;$ moment of inertia about the line through the centroid, parallel to the Y-axis

: $I_ = \;$ moment of inertia about the line through the centroid, parallel to the Z-axis

If the origin of the axis system is positioned at the object's centroid, we can simplify the notation further:

: $I_ = \;$ moment of inertia about the X-axis

: $I_ = \;$ moment of inertia about the Y-axis

: $I_ = \;$ moment of inertia about the Z-axis

Application of moment of inertia

A common equation which describes the relationship between the linear force applied to an object, the object's mass, and the object's linear acceleration, in a frictionless setting, is:

: $F = ma\,$

A similar equation can be used to describes the relationship between the rotational force (torque) applied to an object, the object's rotational mass (moment of inertia), and the object's rotational (angular) acceleration, in a frictionless setting:

: $T = I\,$

Where:

: $T = \,$ torque

: $I = \,$ moment of inertia

: $= \,$ rotational (angular) acceleration

Inertia tensor
The moment of inertia can be used to describe the amount of angular momentum a rigid body possesses, via the relation:

: $\vec = I \vec\,$

For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar.

However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity. The definition of the moment of inertia tensor is very similar to that above, except that it is now expressed as a matrix:

: $I = \sum_i m_i r_i^2 (E-P_i)$

where

:E is the identity matrix
:P is the projection operator.

Alternatively the elements of the inertia tensor can be expressed as:

: $I_ = \sum_i m_i (y_i^2+z_i^2)$
: $I_ = \sum_i m_i (x_i^2+z_i^2)$
: $I_ = \sum_i m_i (x_i^2+y_i^2)$
: $I_ = I_ = -\sum_i m_i x_i y_i\;$
: $I_ = I_ = -\sum_i m_i x_i z_i\;$
: $I_ = I_ = -\sum_i m_i y_i z_i\;$

It is notable that (because it is symmetric), it is always possible to diagonalize the inertia tensor to find the principal axes of the rigid body, those which satisfy the eigenvalue problem:

: $I\vec=\lambda \vec.\,$

In the case of rotation about a principal axis with constant angular velocity, the angular momentum is, like the angular velocity, along this axis, so it r}