Explain angular velocity and angular acceleration about fixed axis and derive the equation of rotational motion and write the analogy between the equations of linear motion and rotational motion.

Explanation

As shown in the figure a rigid body rotating about fixed Z-axis in XYZ cartesion co-ordinate system.
Any particle P of the body circulating in XY-plane.
The angular position of this particle P at `t=0` time is `theta_(0)` and `t=t` time it is `theta_(0)+theta`
`therefore` In time t its angular displacement is `theta`.
Now select X. and Y. parallel to X and Y. Z-axis is already fixed.
Intrateneous angular velocity
`omega` = time rate of change of angular displacement
`therefore omega=(d theta)/(dt)`, is in the direction of fixed Z-axis so it can be taken as scaler.
Angular acceleration `alpha` = time rate of change of angular velocity.
`alpha=(domega)/(dt)`, is in the direction of fixed axis so it can be also taken as scalar.
Equations of pure linear motion
`v=v_(0)+at`
`x=x_(0)+v_(0)t+(1)/(2)at^(2)`
`v^(2)=v_(0)^(2)+2a(x-x_(0))`
where `x_(0)` = initial position
x = final position
`v_(0)` = initial velocity
v = final velocity
a = acceleration, t = time
Equation of pure rotational motion
`omega=omega_(0)+at`
`theta=theta_(0)+omega_(0)t+(1)/(2)alphat^(2)`
`omega^(2)=omega_(0)^(2)+2alpha(theta-theta_(0))`
where `theta_(0)` = Initial angular position
`theta` = Final angular position
`omega_(0)` = Initial angular velocity
`omega` = Final angular velocity
`alpha` = Angular acceleration
t = time