For more details of this early work, see Alpha particle # History of discovery and use.

For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

( For one particle, J = L + S.

For example, it also predicts the variation of the particle density in a gravitational field with height, if.

For example, the probability that an alpha particle striking a beryllium target will produce a neutron can be expressed as the equivalent cross section of beryllium for this type of reaction.

For example, if we suppose at t = 0 the extension is a unit distance ( x = 1 ), and the particle is not moving ( dx / dt = 0 ).

For mathematical purposes, elementary particles are normally treated as point particles, although some particle theories such as string theory posit a physical dimension.

: For an elementary introduction to how Lie algebras are related to particle physics, see the article Particle physics and representation theory.

The red dot overtakes two green dots when moving from the left to the right of the figure. New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front. For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.

For a free non-relativistic particle it follows that

For non-interacting particles, i. e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle, that is

For instance, if a particle is in a state | ψ ⟩, the probability of finding it in a region of volume d < sup > 3 </ sup > x surrounding some position x is

For a single particle, this quantity is the rest mass ; for a system of bound or unbound particles, this quantity is the invariant mass.

For example, in a collision of two particles a reference frame can be chosen where one particle begins at rest.

For example, one can talk about motion of a wave or a quantum particle ( or any other field ) where the configuration consists of probabilities of occupying specific positions.

For example, mā ( high level pitch ) means " mom ", má ( rising pitch ) means " hemp ", mǎ ( falling then rising ) means " horse ", mà ( falling ) means " scold ", and ma ( neutral tone ) is an interrogative particle.

For example, a single photon may be refracted by a lens or exhibit wave interference with itself, but also act as a particle giving a definite result when its position is measured.

The red dot overtakes two green dots when moving from the left to the right of the figure. New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front. For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.

For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x < sub > 0 </ sub > ( neither an eigenstate of position nor of momentum ).

For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus ( Fig.

For example, consider a free particle.

For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators and.

For example, the 3-particle state with one particle in state and two in state is

For the case of light alpha particles scattering off heavy nuclei, as in the experiment performed by Rutherford, the reduced mass is essentially the mass of the alpha particle and the nucleus off of which it scatters is essentially stationary in the lab frame.

For head on collisions between alpha particles and the nucleus, all the kinetic energy of the alpha particle is turned into potential energy and the particle is at rest.

For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.

: For an example of the simple algorithm " Add m + n " described in all three levels see Algorithm examples.

For example, in the first editions of the collection The Mysterious Mr Quin ( 1930 ), in the short story " The Soul of the Croupier ," she described " Hebraic men with hook-noses wearing rather flamboyant jewellery "; in later editions the passage was edited to describe " sallow men " wearing same.

For many years, the events described in " The Curse of Akkad " were thought, like the details of Sargon's birth, to be purely fictional.

For example, a poker hand can be described as a 5-combination ( k = 5 ) of cards from a 52 card deck ( n = 52 ).

For this reason, the history of Cairo during Ottoman times is often described as inconsequential, especially in comparison to other time periods.

For his technical developments, promotion of dairy hygiene and unremunerated propagation of modern cheese-making techniques he has been described as the father of Cheddar cheese.

For each fellowship, anyone who publicly assents to the doctrines described in the statement and is in good standing in their " home ecclesia " is generally welcome to participate in the activities of any other ecclesia.

For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 4.

For Derrida, Genesis and Structure are both inescapable modes of description, there are some things that " must be described in terms of structure, and others which must be described in terms of genesis ," but these two modes of description are difficult to reconcile and this is the tension of the structural problematic.

For example, a hanging tom 12 " in diameter and 8 " deep would be described by Tama as 8x12, but by Pearl as 12x8, and a standard diameter Ludwig snare drum 5 " deep is a 5x14, while a Premier of the same dimensions is a 14x5.

For example, Haeckel described and named hypothetical ancestral microorganisms that have never been found.

For example, if the temperature today, T, is 20 degrees higher than the temperature yesterday, Y, then the problem can be described algebraically as.

For example, the relationship between the circumference, c, and diameter, d, of a circle is described by.

For example, other factors such as modeling ( as described by the social cognitive theory of morality ) are required to explain bullying.

For an economic model described by more than one equation, simultaneous-equation methods may be used to remedy similar problems, including two IV variants, Two-Stage Least Squares ( 2SLS ), and Three-Stage Least Squares ( 3SLS ).

This is how Wright described, in his autobiography, the influence of these exercises on his approach to design: " For several years I sat at the little Kindergarten table-top.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

For clarity, all jumps will be described for a skater jumping counter-clockwise.

For a single degree of freedom oscillator, a system in which the motion can be described by a single coordinate, the natural frequency depends on two system properties: mass and stiffness ; ( providing the system is undamped ).

For example, former field hands ( during the early days of Reconstruction ) are described behaving " as creatures of small intelligence might naturally be expected to do.

For general matrices, Gaussian elimination is usually considered to be stable in practice if you use partial pivoting as described below, even though there are examples for which it is unstable.

For instance, German-speakers more often described, ( f .) " bridge " with words like ' beautiful ', ' elegant ', ' fragile ', ' peaceful ', ' pretty ', and ' slender ', whereas Spanish-speakers, which use puente ( m .) used terms like ' big ', ' dangerous ', ' long ', ' strong ', ' sturdy ', and ' towering '.

For example Hecataeus of Miletus, a geographer and historian in ancient Greece, described inhabitants of the ancient world as well as physical features.