Question

Obtain equation of angular velocity as a function of time for rotating bodies with constant angular acceleration from the first principles.

Answer 1

Hint:We should know angular velocity, angular acceleration.We should have an idea of the relationship between angular acceleration and angular velocity.
Formula Used:
\[\alpha = \dfrac{{d\omega }}{{dt}}\]\[(\alpha {\text{ }} = \] angular acceleration, \[\omega {\text{ }} = {\text{ }}angular{\text{ }}velocity)\]

Complete step by step answer:
Every particle in a rotating body moves in a circular direction. Angular displacement of a given particle about its centre per time is defined as angular velocity. In a pure rotational motion, all parts of a moving body have the same angular velocity.

It is a vector quantity and is described as the rate of change of angular displacement equals the angular speed or rotational speed of an object and the axis about which the object is rotating. The amount of change of angular displacement of the particle at a given period of time is called angular velocity.

In the question we know that the angular acceleration is uniform or constant,
Hence,
\[\alpha = \dfrac{{d\omega }}{{dt}}\]\[ = \] a constant.................... (i)
 We need to get the equation of angular velocity in terms of time. So we need to integrate this equation with respect to time (t),
 \[d\omega = \alpha \cdot dt\]
\[\begin{gathered}
\int {d\omega } = \int {\alpha \cdot dt} \\ \omega = \alpha t + C \\
\end{gathered} \]
 Here α is constant.
 For finding integrating constants, we have to give initial conditions.
So at
\[t = 0\] ,\[\omega = {\omega _0}\],
\[\begin{gathered}
{\omega _0} = 0 + C \\
C = {\omega _0} \\
\end{gathered} \]
Therefore,
\[\omega = \alpha t + {\omega _0}\]
 This is the angular velocity with respect to time at constant acceleration.

Note:Sometimes during the integration the students take w as a constant please always be careful about it because w is the function of time.Hence,it is variable and we can not take it as a constant.