Rotational kinematic equations Knowledge-Based Study

Study notes

1. Angular Kinematic Quantities

   Rotational motion describes an object rotating around a fixed axis. Just as linear motion is described by position, velocity, and acceleration, rotational motion is described by analogous angular quantities:

   • Angular Position ($\theta$): The angle measured in radians (rad) relative to a reference direction.
   • Angular Displacement ($\Delta \theta$): The change in angular position, given by $\Delta \theta = \theta_f - \theta_i$.
   • Angular Velocity ($\omega$): The rate of change of angular position with respect to time, measured in radians per second (rad/s):
     $$\omega = \frac{d\theta}{dt}$$
   • Angular Acceleration ($\alpha$): The rate of change of angular velocity with respect to time, measured in radians per second squared (rad/s$^2$):
     $$\alpha = \frac{d\omega}{dt}$$

   Rotational Motion with Constant $\alpha$

   When an object undergoes rotation with constant angular acceleration, the mathematical relationships mirror the standard linear kinematic equations. The four foundational rotational kinematic equations are:

   1. Velocity-Time Relation:
      $$\omega_f = \omega_i + \alpha t$$
   2. Displacement-Time Relation:
      $$\Delta \theta = \omega_i t + \frac{1}{2}\alpha t^2$$
   3. Velocity-Displacement Relation (Independent of Time):
      $$\omega_f^2 = \omega_i^2 + 2\alpha \Delta \theta$$
   4. Average Velocity Relation:
      $$\Delta \theta = \left(\frac{\omega_i + \omega_f}{2}\right)t$$

   Note: These formulas are valid only when the angular acceleration $\alpha$ is uniform across the entire time interval.

   Linear versus Rotational Correspondence

   Comparing linear and angular variables helps solidify the application of these formulas:

   • Linear displacement $\Delta x$ corresponds directly to angular displacement $\Delta \theta$.
   • Initial velocity $v_i$ corresponds to initial angular velocity $\omega_i$.
   • Final velocity $v_f$ corresponds to final angular velocity $\omega_f$.
   • Linear acceleration $a$ corresponds to angular acceleration $\alpha$.
   • Time $t$ is an identical parameter in both systems.

   For a point at a fixed radius $r$ from the axis of rotation, linear and rotational values are linked by the relations $s = r\theta$, $v_t = r\omega$, and $a_t = r\alpha$, where $v_t$ and $a_t$ represent tangential velocity and tangential acceleration, respectively.

   Solving Rotational Kinematics Problems

   To solve problems involving rotational kinematic equations, follow these structured steps:

   1. Identify the known values and state the target unknown variable.
   2. Ensure all units are converted to standard SI units (convert revolutions per minute or degrees to radians).
   3. Select the kinematic equation containing only the knowns and the single desired unknown.

   Example Walkthrough

   Problem: A flywheel accelerates uniformly from rest to an angular speed of 12 rad/s in 4.0 s. Calculate its total angular displacement during this interval.

   • Given: $\omega_i = 0\text{ rad/s}$, $\omega_f = 12\text{ rad/s}$, $t = 4.0\text{ s}$
   • Target: $\Delta \theta$
   • Equation: $\Delta \theta = \left(\frac{\omega_i + \omega_f}{2}\right)t$
   • Calculation:
     $$\Delta \theta = \left(\frac{0 + 12}{2}\right)(4.0) = (6)(4.0) = 24\text{ rad}$$

   The flywheel rotates through an angle of 24 radians.