Radian
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Some common angles, measured in radians.The radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.2 rad" or "1.2c" (the second symbol can be mistaken for a degree: "1.2°").
However, the radian is the de facto unit of angular measurement for mathematicians, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol ° is used.
The radian was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit. The SI unit of solid angle measurement is the steradian.
Contents [hide]
1 Definition
2 History
3 Conversions
3.1 Conversion between radians and degrees
3.2 Conversion between radians and grads
4 Reasons why radians are preferred in mathematics
5 Dimensional analysis
6 Use in physics
7 SI multiples
8 References
9 See also
10 External links
[edit] Definition
An angle of 1 radian subtends an arc equal in length to the radius of the circle.One radian is the angle subtended at the center of a circle by an arc of circumference that is equal in length to the radius of the circle.
More generally, the magnitude in radians of any angle subtended by two radii is equal to the ratio of the length of the enclosed arc to the radius of the circle; that is, θ = s/r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ.
It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr/r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.
[edit] History
The concept of a radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714.[1] He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure.
The term radian first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was a brother of Lord Kelvin. He used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, hesitated between rad, radial and radian. In 1874, Muir adopted radian after a consultation with James Thomson.[2][3][4]
[edit] Conversions
[edit] Conversion between radians and degrees
As explained above under "Definition", one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π. For example,
Conversely, to convert from degrees to radians, multiply by π/180. For example,
The table shows the conversion of some common angles.
[edit] Conversion between radians and grads
2π radians are equal to one complete revolution, which is 400g. So, to convert from radians to grads multiply by 200/π, and to convert from grads to radians multiply by π/200. For example,
[edit] Reasons why radians are preferred in mathematics
In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. One important reason is that results involving trigonometric functions are simple and "natural" when the function's argument is expressed in radians. For example, the use of radians leads to the simple identity
,
which is the basis of many other elegant identities in mathematics, including
.
The trigonometric functions also have simpler series expansions when radians are used; for example, the following the Taylor series for sin x:
If x were expressed in degrees then the series would contain messy factors involving powers of π/180.
[edit] Dimensional analysis
Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless.
Another way to see the dimensionlessness of the radian is in the series representations of the trigonometric functions, such as the Taylor series for sin x mentioned earlier:
If x had units, then the sum would be meaningless: the linear term x cannot be added to (or have subtracted) the cubic term x3 / 3!. Therefore, x must be dimensionless.
[edit] Use in physics
The radian is widely used in physics when angular measurements are required. For example, angular velocity is typically measured in radians per second (rad/s). One revolution per second is equal to 2π radians per second.
Similarly, angular acceleration is often measured in radians per second per second (rad/s2).
[edit] SI multiples
SI prefixes have limited use with radians. The milliradian (0.001 rad, or 1 mrad) is used in gunnery and general targeting, because it corresponds to 1 m at a range of 1000 m (at such small angles, the curvature can be considered negligible). The divergence of laser beams is also usually measured in milliradians. Smaller units, like microradians (μrads) and nanoradians (nrads) are used in astronomy, and can also be used to measure the beam quality of lasers with ultra-low divergence. Similarly, the prefixes smaller than milli- are potentially useful in measuring extremely small angles. However, the larger prefixes have no apparent utility, mainly because to exceed 2π radians is to begin the same circle (or revolutionary cycle) again.
[edit] References
^ Biography of Roger Cotes, The MacTutor History of Mathematics
^ Florian Cajori, 1929, History of Mathematical Notations, Vol. 2, pp. 147–148
^ Nature, 1910, Vol. 83, pp. 156, 217, and 459–460
^ Earliest Known Uses of Some of the Words of Mathematics
[edit] See also
Trigonometry
Harmonic analysis
Angular frequency
Grad
Degree
Steradian - the "square radian"
[edit] External links
Look up radian in Wiktionary, the free dictionary.Radian at MathWorld
Radians/Degree Converter
Retrieved from "http://en.wikipedia.org/wiki/Radian"
Great discussion going on over here....All have made such great points...now the hard part is translating it those students who have downstroke "blackout"
Pace in my mind is the direct relationship of the Hand Speed and the Surface Speed..I could be wrong and would stand to be corrected
Arc Velocity is surface speed
Arc Accleration is the result of the radius changing from smaller to larger
Angular Velocity is the rate of rotation of all components and their respective centers...in this case the radius in relation to clubhead travel controlled by #3 , Hand travel and turning rate of the pivot components
Angular Acceleration is the change in hand speed .....via the pivot or accumulators #4 and #1
Rhythm is the RPM roll of #3 and the turning rate of the pivot to maintain the same Angular velcocities or the rate of closure of the clubface, travel of the orbitng clubhead, to the selcted plane angle
Aside from the Wiki Tiki protractor pocket protector pollution, I really like this. Without looking any Wiki's up, I'll bite.
Pace -
I might agree that it's a measure of speed over distance and can be applied to both hands and clubhead as long as the two are measured in relation to one another. I think there is a lot of meat on that bone.
Arc Velocity - Are we talking from the top/end to follow through only? That would seem to be the a clubhead surface speed MPH measurement only. I had defined pace in this respect. Are you putting hands here as well?
Arc Acceleration - That would be a measurement of the overtaking rate of the clubhead from release to impact. The result may indicate pulley size, but only in swinging. How would this differ from your view of Pace?
Angular Velocity - Per your definition, I think you need to measure RPM, not speed. If you measure speed, you need a reference point. Angular Velocity by name would be a speed measurement of a linear object (hands?) in relation (from a golfing perspective) to a circle. Just a wild guess.
Angular Acceleration - Rate of change in hand speed. Not sure that it would relate to anything else.
OK Annikan - now that I've butchered that, what is my homework assignment?
I have a feeling I'm about to get a low inside pitch straight at the knees.
__________________
Bagger
1-H "Because of questions of all kinds, reams of additional detail must be made available - but separately, and probably endlessly." Homer Kelly
Arc Velocity is a measurement along the circumferece(travel distance)
Angular Velocity is a measurement around an axis of rotation(instantaneous centers)
I will go through my notes from class (ExSC 570 - Biomechanics for Educators - Dr. Brian Bergeman here at Campbell University) to see if I'm correct and will post them later
mmmkay - I'll do my homework because I have do have my decoder ring and the message is loud and clear.
Watch out for wedgies.
Homework Assignment #1 - Research Angular Kinematics...You can use Wiki Tiki if you like or "Biomechanics of Sports and Excercise" by Peter M. McGinnis...Make sure to turn to page 153 and read the definition of angular velocity...Hint...PBH beat you to it!!!
The angular velocity vector points up for counterclockwise rotation and down for clockwise rotation, as specified by the right-hand rule.Rotational motion is the description of the turning of an object and involves the following three quantities, as do linear motion:
Angular displacement: Angular position θ is the angle that a line from the axis of rotation to a point on an object makes with respect to the positive x-axis, which is measured counterclockwise.
Angular velocity: The magnitude of the angular velocity ω is the rate at which the angular position θ changes with respect to time t:
Angular acceleration: The magnitude of the angular acceleration α is the rate at which the angular velocity ω changes with respect to time t:
Last edited by annikan skywalker : 06-12-2007 at 05:58 PM.
Homework Assignment #1 - Research Angular Kinematics...You can use Wiki Tiki if you like or "Biomechanics of Sports and Excercise" by Peter M. McGinnis...Make sure to turn to page 153 and read the definition of angular velocity...
I promise to do the homework, but this sure feels like punishment for abusing your N.C. buddy.
Do you promise light bulbs are gunna start pop'in like John Coffey made em in The Green Mile?
Or are you leading me down the hallway to visit "ole sparky"?
I gotsta know boss.
__________________
Bagger
1-H "Because of questions of all kinds, reams of additional detail must be made available - but separately, and probably endlessly." Homer Kelly
1) Angle: (Here) the angle between the golf shaft at the moment and some vector on the inclined plane.
2) Angular velocity: The change rate of angle
3) Angular acceleration: The change rate of angular velocity.
Linear Mode
