Rotational Dynamics: Instantaneous Angular Velocity and Angular Acceleration

Instantaneous Angular Velocity

• Definition and Mathematical Formulation:

  • The instantaneous angular velocity ($\omega$) of a particle or rotating body is defined as the limit of the ratio of angular displacement to the time interval as the time interval approaches zero.
  • Mathematically, it is expressed as:     $\omega = \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$
  • In this expression:
  • $\theta$ represents the angular displacement.
  • $t$ represents time.
  • $\Delta \theta$ represents the change in angular displacement during the time interval $\Delta t$.
  • $\frac{d\theta}{dt}$ represents the first derivative of angular displacement with respect to time.

• Units of Measurement:

  • The standard SI unit of angular velocity is radian per second ($\text{radian / sec}$ or $\text{rad/s}$).

• Relationship with Time Period ($T$):

  • For one complete circular rotation, the angular displacement $\Delta \theta$ equals $2\pi\,\text{radians}$.
  • The time required to complete one full revolution is defined as the time period $T$ (where $\Delta t = T$).
  • Substituting these values into the formula yields:     $\omega = \frac{2\pi}{T}$

• Relationship with Frequency ($n$):

  • Frequency ($n$) is defined as the reciprocal of the time period:     $n = \frac{1}{T}$
  • Expressing angular velocity in terms of frequency gives:     $\omega = 2\pi n$

Angular Acceleration

• Condition for Angular Acceleration:

  • If the angular velocity of a rotating body is not constant over time, the body experiences an angular acceleration.

• Average Angular Acceleration:

  • Let $\omega_1$ and $\omega_2$ be the instantaneous angular velocities of the rotating body at time $t_1$ and time $t_2$, respectively.
  • The average angular acceleration ($\alpha_{\text{avg}}$) of the rotating body in the time interval from $t_1$ to $t_2$ is defined as the change in angular velocity divided by the time interval:     $\alpha_{\text{avg}} = \frac{\omega_2 - \omega_1}{t_2 - t_1} = \frac{\Delta \omega}{\Delta t}$
  • Here, $\Delta \omega$ represents the change in angular velocity that occurs during the time interval $\Delta t$.

• Instantaneous Angular Acceleration:

  • Instantaneous angular acceleration ($\alpha$) is the angular acceleration of the body at a specific instant in time.
  • It is defined as the limiting value of the average angular acceleration as the time interval $\Delta t$ approaches zero:     $\alpha = \lim_{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt}$
  • Expressed as a second derivative of angular displacement with respect to time:     $\alpha = \frac{d^2\theta}{dt^2}$