View source for Tangential speed

{{Short description|How quickly an object undergoes movement in a circular path.}}
[[File:Angular-and-tangential-speed.svg|thumb|upright=1.5|Tangential speed ({{math|''v''}}) and [[angular speed]] ({{math|''ω''}}) on a spinning disc of radius {{math|''r''}}.]]
{{Classical mechanics|rotational}}

'''Tangential speed''' is the [[speed]] of an object undergoing [[circular motion]], i.e., moving along a [[circular path]].<ref name="Hewitt 2007, p. 131">{{harvnb|Hewitt|2007|p=131}}</ref> A point on the outside edge of a [[merry-go-round]] or [[turntable]] travels a greater distance in one complete [[rotation (unit)|rotation]] than a point nearer the center. Travelling a greater distance in the same time means a greater speed, and so linear speed is greater on the outer edge of a rotating object than it is closer to the axis. This speed along a circular path is known as ''tangential speed'' because the direction of motion is [[Tangent lines to circles|tangent]] to the [[circumference]] of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, and is measured in [[SI units]] as meters per second (m/s).

==Formula==
[[Rotational speed]] (or rotational frequency) measures the number of revolutions per unit of time. All parts of a rigid merry-go-round or turntable turn about the axis of rotation in the same amount of time. Thus, all parts share the same rate of rotation, or the same number of rotations or revolutions per unit of time.
When a direction is assigned to rotational speed, it is known as [[rotational velocity]], a vector whose magnitude is the rotational speed.
([[Angular speed]] and [[angular velocity]] are related to the rotational speed and velocity by a factor of 2{{pi}}, the number of [[radians]] turned in a full rotation.) 

Tangential speed and rotational speed are related: the faster an object rotates around an axis, the larger the speed. Tangential speed is directly proportional to rotational speed at any fixed distance from the axis of rotation.<ref name="Hewitt 2007, p. 131"/> However, tangential speed, unlike rotational speed, depends on radial distance (the distance from the axis). For a platform rotating with a fixed rotational speed, the tangential speed in the centre is zero. Towards the edge of the platform the tangential speed increases proportional to the distance from the axis.<ref>{{harvnb|Hewitt|2007|p=132}}</ref> In equation form:
<math display="block">v \propto \!\, r \omega\,,</math>

where {{math|''v''}} is tangential speed and {{math|''ω''}} (Greek letter [[omega]]) is rotational speed. One moves faster if the rate of rotation increases (a larger value for {{math|''ω''}}), and one also moves faster if movement farther from the axis occurs (a larger value for {{math|''r''}}). Move twice as far from the rotational axis at the centre and you move twice as fast. Move out three times as far, and you have three times as much tangential speed. In any kind of rotating system, tangential speed depends on how far you are from the axis of rotation.

When proper units are used for tangential speed {{math|''v''}}, rotational speed {{math|''ω''}}, and radial distance {{math|''r''}}, the direct proportion of {{math|''v''}} to both {{math|''r''}} and {{math|''ω''}} becomes the exact equation
<math display="block">v = r\omega\,.</math>This comes from the following: the linear (tangential) velocity of an object in rotation is the rate at which it covers the circumference's length:

:<math>v = \frac{2\pi r}{T}</math>

The angular velocity <math display="inline">\omega</math> is defined as <math>2\pi /T</math>, where ''T'' is the [[rotation period (physics)|rotation period]], hence <math>v = \omega r</math>.

Thus, tangential speed will be directly proportional to {{math|''r''}} when all parts of a system simultaneously have the same {{math|''ω''}}, as for a wheel, disk, or rigid wand.

==References==
{{Reflist}}
{{Refbegin}}
* {{cite book | last=Hewitt | first=P.G. | title=Conceptual Physics | publisher=Pearson Education | year=2007 | isbn=978-81-317-1553-6 | url=https://books.google.com/books?id=A5HylF_5tLEC | access-date=2023-07-20}}
* [[Richard Feynman|Richard P. Feynman]], Robert B. Leighton, Matthew Sands. [[The Feynman Lectures on Physics]], Volume I, [https://feynmanlectures.caltech.edu/I_08.html#Ch8-S2 Section 8–2]. [[Addison-Wesley]], Reading, Massachusetts (1963). {{ISBN|0-201-02116-1}}.
{{Refend}}

{{Kinematics}}
{{Classical mechanics derived SI units}}
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[[Category:Physical quantities]]
[[Category:Temporal rates]]
[[Category:Velocity]]