Distances/Angular momenta

This diagram describes the relationship between force (F), torque (τ), momentum (p), and angular momentum (L) vectors in a rotating system. 'r' is the radius. Credit: Yawe.

Angular momenta, or angular momentum, is a lecture from the radiation astronomy department. Usually, such a classical field would be a department of physics lecture.

It is a lecture in a series about distances. Large distances are significant in astronomy.

Referring to the diagram on the right, an angular momentum L of a particle about an origin is given by

$\mathbf {L} =\mathbf {r} \times \mathbf {p}$

where r is the radius vector of the particle relative to the origin, p is the linear momentum of the particle, and × denotes the cross product (r · p sin θ). Theta is the angle between r and p.

The radius vector consists of two parts or concepts: a distance, or displacement, and a direction, indicated by the arrow.

Theoretical classical mechanics

Def. an "[amount of] intervening space between two points, usually geographical points, usually (but not necessarily) measured along a straight line" is called a distance.

Def. the "inevitable progression into the future with the passing of present events into the past" or the "inevitable passing of events from future to present then past" is called time.

Def. the "quantity of matter which a body contains, irrespective of its bulk or volume" is called a mass.

Def. the "rate of motion or action, specifically the magnitude of the velocity; the rate distance is traversed in a given time" is called the speed.

Kinetics

The angular velocity of the particle at P with respect to the origin O is determined by the perpendicular component of the velocity vector v. Credit: Krishnavedala.

The angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. The direction of the angular velocity pseudovector is along the axis of rotation; in this case (counter-clockwise rotation) the vector points up. Credit: DnetSvg.

Def. a "quantity [...] cohering together so as to make one body, or an aggregation of particles or things which collectively make one body or quantity" is called a mass.

Mass is an idea.

Def. "objects in motion, but not with the forces involved" is called kinematics, or the science of kinematics.

Def. a "property of a body that resists any change to its uniform motion" is called inertia.

Mass and inertia are generally considered equivalent.

Isaac Newton's laws of motion contain the idea of inertia.

Newton's First law: "Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed."

While motion in a straight line, or rectilinear motion can be produced over limited distances in a laboratory, it may not occur naturally.

Def. "the product of [a body's] mass and velocity" is called momentum.

Def. "the rotary inertia of a system [such as] an isolated rigid body [...] is a measure of the extent to which an object will continue to rotate in the absence of an applied torque" is called angular momentum.

Def. a "rotational or twisting effect of a force" is called a torque.

Def. a "turning effect of a force applied to a rotational system at a distance from the axis of rotation" is called a moment of force.

"The moment is equal to the magnitude of the force multiplied by the perpendicular distance between its line of action and the axis of rotation."

A torque and a moment of force are the same. Each is a "unit of work done, or energy expended".

Def. "the effects of forces on moving bodies" is called kinetics, or the science of kinetics.

Moment of inertia

The moment of inertia and angular momenta are different for every possible configuration of mass and axis of rotation. Credit: PanCiasteczko.

"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:

$\mathbf {L} =I{\boldsymbol {\omega }}$

where I is the moment of inertia of the object (in general, a tensor quantity), and ω is the angular velocity.

The moment of inertia is the mass property of a rigid body that defines the torque needed for a desired angular acceleration about an axis of rotation. Moment of inertia depends on the shape of the body and may be different around different axes of rotation. A larger moment of inertia around a given axis requires more torque to increase the rotation, or to stop the rotation, of a body about that axis. Moment of inertia depends on the amount and distribution of its mass, and can be found through the sum of moments of inertia of the masses making up the whole object, under the same conditions.

Angular velocity

In two dimensions the angular velocity ω is given by

$\omega ={\frac {d\phi }{dt}}$

This is related to the cross-radial (tangential) velocity by:

$\mathrm {v} _{\perp }=r\,{\frac {d\phi }{dt}}$

An explicit formula for v_⊥ in terms of v and θ is:

$\mathrm {v} _{\perp }=|\mathrm {\mathbf {v} } |\,\sin(\theta )$

Combining the above equations gives a formula for ω:

$\omega ={\frac {|\mathrm {\mathbf {v} } |\sin(\theta )}{|\mathrm {\mathbf {r} } |}}$

Conservation of angular momentum

A figure skater conserves angular momentum – her angular rotational speed increases as her moment of inertia decreases by drawing in her arms and legs. Credit: Deerstop.

"In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque."

Angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved).

"A change in angular momentum is proportional to the applied torque and occurs about the same axis as that torque."

Requiring the system to be closed is equivalent to requiring that no external influence, in the form of a torque, acts upon it.

"A body continues in a state of rest or of uniform rotation unless compelled by a torque to change its state."

With no external influence to act upon it, the original angular momentum of the system is conserved.

Orbital mechanics

A massless (or per unit mass) angular momentum is defined by

$\mathbf {h} =\mathbf {r} \times \mathbf {v} ,$

called specific angular momentum, where $\mathbf {L} =m\mathbf {h} .$

Earth system

Main resources: Planets/Earth/System, Earth system, and Earth and Moon

File:Retroreflector array.jpg

This is a photograph of a retroreflector array placed by the crew of Apollo 14 on the lunar surface. Credit: Alan B. Shepard, Jr., and Edgar D. Mitchell, during EVA 1 of Apollo 14 on the Moon.

File:Geographical distribution retroreflectors.jpg

Plotted are the geographical distribution of the retroreflector arrays on the lunar surface. Credit: J. O. Dickey, P. L. Bender, J. E. Faller, X X Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veilet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder.

"On 21 July 1969, during the first manned lunar mission, Apollo 11, the first retroreflector was placed on the moon, enabling highly accurate measurements of the Earth - moon separation by means of laser ranging."

"The locations of the three Apollo [A-11, A-14, and A-15] arrays plus one French-built array still operating on the Soviet roving vehicle Lunakhod 2 [L-1 and L-2 are shown in the image on the left and] provide a favorable geometry for studying the rotations of the moon and for separating these rotations from lunar orbital motion and geodynamic effects [...]."

"Lunar laser ranging consists of measuring the round-trip travel time and thus the separation between transmitter and reflector."

"Retroreflector arrays provide optical points on the moon toward which one can fire a laser pulse and receive back a localized and recognizable signal. Ranging accuracies on the order of a centimeter are immediately possible if one has sufficiently short laser pulse lengths with high power."

Although an "order-of-magnitude improvement in accuracy [has occurred since the Apollo program], the early data are still important in the separation of effects with long characteristic timescales, notably precession, nutation, relativistic geodetic precession, tidal acceleration, the primary lunar oblateness term (J₂), and the relative orientation of the planes of the Earth's equator, the lunar orbit, and the ecliptic."

"The data set considered here consists of over 8300 normal-point ranges (8) spanning the period between August 1969 and December 1993; the observatories and the lunar reflectors included in the analysis are listed in Table 1. The data are analyzed with a model that calculates the light travel time between the observatory and the reflector, accounting for the orientation of the Earth and moon, the distance between the centers of the two bodies, solid tides on both bodies, plate motion, atmospheric delay, and relativity (13). The fitted parameters include the geocentric locations of the observatories; corrections to the variation of latitude (that is, polar motion); the orbit of the moon about the Earth; the Earth's obliquity, precession, and nutation; plus lunar parameters including the selenocentric reflector coordinates, fractional moment-of-inertia differences, gravitational third-degree harmonics, a lunar Love number, and rotational dissipation."

"The mean Earth-moon distance is 385,000 km; the radii of the Earth and moon are 6371 and 1738 km, respectively."

"The moon's orbit is strongly distorted from a simple elliptical path by the solar attraction-the instantaneous eccentricity varies by a factor of 2 (0.03 to 0.07)."

"[A]ccuracies are degraded when extrapolated outside the span of observations."

"The two largest solar perturbations in distance r [the distance between the centers of the Earth and moon] are 3699 km (monthly) and 2956 km (semimonthly)."

Hypotheses

Main resource: Hypotheses

1. To use angular momentum or energy a mass must be assigned.

See also

• Angular momentum and energy
• Angular momenta quiz