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Tangential speed is thespeed of an object undergoingcircular motion, i.e., moving along acircular path.^[1] A point on the outside edge of amerry-go-round orturntable travels a greater distance in one completerotation 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 astangential speed because the direction of motion istangent to thecircumference of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, and is measured inSI units as meters per second (m/s).

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 asrotational velocity, a vector whose magnitude is the rotational speed.(Angular speed andangular velocity are related to the rotational speed and velocity by a factor of 2π, the number ofradians 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.^[1] 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.^[2] In equation form:$${\displaystyle v\propto r\omega ,}$$

wherev is tangential speed andω (Greek letteromega) is rotational speed. One moves faster if the rate of rotation increases (a larger value forω), and one also moves faster if movement farther from the axis occurs (a larger value forr). 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 speedv, rotational speedω, and radial distancer, the direct proportion ofv to bothr andω becomes the exact equation$${\displaystyle v=r\omega .}$$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:

${\displaystyle v=\frac{2\pi r}{T}}$

The angular velocity$\omega$ is defined as${\displaystyle 2\pi /T}$, whereT is therotation period, hence${\displaystyle v=\omega r}$.

Thus, tangential speed will be directly proportional tor when all parts of a system simultaneously have the sameω, as for a wheel, disk, or rigid wand.

For tangentialvelocity vector (rapidity or speed is his norm or module) is thevector product:$${\displaystyle \overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}=||\overrightarrow{\omega }||||\overrightarrow{r}||\sin (|\mathrm{\Delta }\theta |)\cdot {\hat{u}}_n=\overrightarrow{v}=\dot{\overrightarrow{r}}=\frac{d\overrightarrow{r}}{dt}}$$Because of theright hand rule linear tangential velocity vector points tangential to the rotation.

Where${\displaystyle \overrightarrow{\omega }=\frac{d\beta }{dt}{\hat{u}}_{\beta }}$ is the angular velocity (angular frequency) vector normal to the plane of rotation of the body, where${\displaystyle \beta }$ is the angle (scalar in radians) of the rotational movement (similar to r that is the norm (scalar) of the translational movement position vector), measured in rad./s = 1/s because rad. radians are adimensional.

${\displaystyle \overrightarrow{r}}$ is theposition vector (equivalent to radio) to the rotating puntual particle or distributedcontinuous body or where is measured the tangential velocity in a body, measured in meters m.

${\displaystyle {\hat{u}}_n}$ is the normal (to the plane of${\displaystyle \overrightarrow{\omega }}$ and${\displaystyle \overrightarrow{r}}$)unit vector.

${\displaystyle \theta }$ are the angles of the vectors${\displaystyle \overrightarrow{\omega }}$ and${\displaystyle \overrightarrow{r}}$ in their common plane where they are, form or describe.

Rapidity or speed${\displaystyle v}$ is thenorm or module of velocity vector${\displaystyle \overrightarrow{v}}$:

${\displaystyle v=||\overrightarrow{v}||=||\overrightarrow{\omega }\times \overrightarrow{r}||=||\overrightarrow{\omega }||||\overrightarrow{r}||\sin (|\mathrm{\Delta }\theta |)=v}$

${\displaystyle v=||\overrightarrow{v}||=||\overrightarrow{\omega }\times \overrightarrow{r}||=||\overrightarrow{\omega }||||\overrightarrow{r}||=\omega r=v}$

Only if:${\displaystyle \sin (|\mathrm{\Delta }\theta |)=1}$, when:${\displaystyle |\mathrm{\Delta }\theta |=|{\theta }_{\overrightarrow{r}}-{\theta }_{\overrightarrow{\omega }}|=\frac{\pi }{2}={90}^o}$, when:${\displaystyle \overrightarrow{\omega }\perp \overrightarrow{r}}$ which means that angular velocity vector is orthogonal (perpendicular) to the position vector.

Tangential acceleration${\displaystyle \overrightarrow{a}}$ is:

${\displaystyle \overrightarrow{a}=\overrightarrow{\omega }\times (\overrightarrow{\omega }\times \overrightarrow{r})=\ddot{\overrightarrow{r}}}$ , =[m/s²]

which means that is from ancentripetal force that is then the fictitious force, not the fictitiouscentrifugal force in its opposite direction

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