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, 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 67% rotation. In equation form: v \propto \!\, r \omega\,, where is tangential speed and (Greek letter
omega) is rotational speed. One moves faster if the rate of rotation increases (a larger value for ), and one also moves faster if movement farther from the axis occurs (a larger value for ). 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 , rotational speed , and radial distance , the direct proportion of to both and becomes the exact equation 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: :v = \frac{2\pi r}{T} The angular velocity \omega is defined as 2\pi /T, where
T is the
rotation period, hence v = \omega r. Thus, tangential speed will be directly proportional to when all parts of a system simultaneously have the same , as for a wheel, disk, or rigid wand. For tangential
velocity vector (rapidity or speed is his norm or module) is the
vector product:\vec{v} = \vec{\omega} \times \vec{r}=||\vec{\omega}||||\vec{r}||\sin(|\Delta \theta|)\cdot\hat{u}_n =\vec{v}=\dot{\vec{r}}= {d\vec{r} \over dt} Because of the
right hand rule linear tangential velocity vector points tangential to the rotation. Where \vec{\omega}={d \beta \over dt} \hat{u}_\beta is the angular velocity (angular frequency) vector normal to the plane of rotation of the body, where \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. \vec{r} is the
position vector (equivalent to radius) to the rotating puntual particle or distributed
continuous body or where is measured the tangential velocity in a body, measured in meters m. \hat{u}_n is the normal (to the plane of \vec{\omega} and \vec{r})
unit vector. \theta are the angles of the vectors \vec{\omega} and \vec{r} in their common plane where they are, form or describe. Rapidity or speed v is the
norm or module of velocity vector \vec{v}: v=||\vec{v}|| = ||\vec{\omega} \times \vec{r}|| = ||\vec{\omega}||||\vec{r}|| \sin(|\Delta \theta|)=v v=||\vec{v}|| = ||\vec{\omega} \times \vec{r}|| = ||\vec{\omega}||||\vec{r}||=\omega r=v Only if: \sin(|\Delta \theta|)=1, when: |\Delta \theta|=|\theta_\vec{r}-\theta_\vec{\omega}| = {\pi \over 2}=90^o , when: \vec{\omega} \perp \vec{r} which means that angular velocity vector is orthogonal (perpendicular) to the position vector. Tangential acceleration \vec{a} is: \vec{a}=\vec{\omega}\times ( \vec{\omega } \times \vec{r})=\ddot{\vec{r}} , =[m/s²] which means that is from an
centripetal force that is then the fictitious force, not the fictitious
centrifugal force in its opposite direction ==References==