Consider the following circle rotating about its centre, in the anti-clockwise direction (it is assumed that we are working in the "normal" cartesian coordinate system, i.e. the "right-handed" coordinates).
In talking about the rotational equivalent of the various basic equations, it is useful to summarise the typical notation used in rotational dynamics.
Position or Displacement
The "position" is typically expressed as the angle around the center that the body has travelled, θ. Correspondingly, the "arc distance" travelled or the displacement is often denoted by \(s\).
Angle: \(\theta\)
Arc: \(s\)
It follows from the definition that:
\[s=r\theta\]
Need to be careful about the "direction". Typically, in a cartesian coordinate system, positive displacement of θ takes the body from the x-axis counter-clockwise, as illustrated above. In 3D, rotation is around a specific axis of rotation.
Angular Velocity
Change in angular displacement per unit time is called angular velocity. The symbol for angular velocity is ω and the units are typically rad per second.
\[\omega=\frac{d\theta}{dt}\]
\[\omega=\frac{d\theta}{dt}\]
For a small time interval \(\Delta t\) and angle change \(\Delta \theta\) it can be expressed in this discrete form.
\[\omega = \frac{\Delta \theta}{\Delta t}\]
Angular speed is the magnitude of angular velocity.
In 2D, ω is a scalar.
Angular Acceleration
Change in angular velocity per unit time is called angular acceleration. The symbol for angular velocity is α.
\[\alpha=\frac{d\omega}{dt}\]
The definition implies that angular acceleration is zero if the anuglar velocity is constant. Therefore, a body in uniform circular motion has zero angular aceleration. That does not mean that its linear acceleraiton is zero.
Period, Frequency and Angular Velocity
Period
Time taken by a body performing circular motion to complete one revolution is called the period of revolution or period time, or simply period, denoted by capital letter T.
\[T=\frac{2\pi}{\omega}\]
Frequency
The number of revolutions by the body per unit time is called as frequency of revolution. Frequency is typically denoted by letter f. The unit of frequency is hertz (Hz).
\[f=\frac{1}{T}\]
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