ANGULAR VELOCITY
Angular velocity:The rate of change of angular displacement is called angular velocity.it is denoted by \( \omega \) ,units are rad/sec.
Let the body be rotating in counter clockwise direction.At the time t1,the particle is at P(t1)while t2 at time it is at P(t2):the angular positions are \( \theta _1 \) and \( \theta _2 \) .Therefore the angular displacement is
\( \theta _2 - \theta _1 = \Delta \theta \) during the time interval \( \Delta t \)
So that average angular speed \( \overline \omega \) of the partcile in this time interval will be
\( \overline \omega = \frac{{\Delta \theta }} {{\Delta t}} \)
The instantaneous angular speed is defined as limit to which ratio \( \frac{{\Delta \theta }} {{\Delta t}} \) approaches as \( \Delta t \) tends to zero.i.e.,
\(
\omega = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \theta }}
{{\Delta t}} = \frac{{d\theta }}
{{dt}}
\)
Since in a rigid body interparticle distance is fixed, therefore if one of the particles undergoes an angular displacement \(d\theta\) in time dt so must the others which implies that ratio \(
\frac{{d\theta }}
{{dt}}
\) or angular speed \(\omega\) will be same for every particle in the body.This means that rather than speaking the angular velocity for a single particle we can state for the whole rigid body,e.g., the rigid body rotates with an angular velocity \(\omega\) or it is the characteristics of the body as a whole.Since \(d\theta\) has no dimensions, has the dimensions of an inverse time (T-1).The unit of is commonly taken as rad/sec or rev/sec.
Relation between time period and angular velocity is Time period T=\(
\frac{{2\Pi }}
{\omega }
\)
If is the angular displacement,number of rotation completed=\(
= \frac{\theta }
{{2\Pi }}
\)
Angular velocity of seconds hand= \(
\frac{{2\Pi }}
{{60}}rads^{ - 1}
\)
Angular velocity of mintes hand= \(
\frac{{2\Pi }}
{{60 \times 60}}rad_s^{ - 1}
\)
Angular velocity of hours hand= \(
\frac{{2\Pi }}
{{12 \times 60 \times 60}}rad_s^{ - 1}
\)
Relation between freequency and angular velocity frequency n= \(
\frac{1}
{T} = \frac{\omega }
{{2\Pi }}
\).
Angular velocity of self rotation of the earth= \(
\frac{{2\Pi }}
{{24 \times 60 \times 60}}rad_s^{ - 1}
\)