Rotational Kinematics Notes

Rotational Kinematics vs. Linear Kinematics

Direct Definitions of Linear and Rotational Variables

• Linear position: $x$
• Angular position: $\theta$
• Linear velocity: $\frac{\Delta x}{\Delta t}$
• Angular velocity: $\frac{\Delta \theta}{\Delta t}$
• Linear acceleration: $\frac{\Delta v}{\Delta t}$
• Angular acceleration: $\frac{\Delta \omega}{\Delta t}$

Constant Acceleration Equations

• The forms of constant linear acceleration equations and constant angular acceleration equations are very much the same; only the names are different because the same physics applies to rotation and translation.
• Subscript zero denotes the initial position or velocity.
• The bar denotes the average velocities.
• In a constant angular velocity situation, the equations simplify to the same form as in linear scenarios, with angular acceleration set to zero.
• Constant acceleration, be it linear or angular, leads to the same mathematical forms for the kinematic variables.

Graphical Models

• Graphical models of kinematic variables in rotational form for a constant angular acceleration situation have the exact same shapes as the linear variables.
• Angular position vs. time plot: parabolic in shape; concavity indicates positive or negative angular acceleration.
• Area under the angular velocity vs. time graph: total angular displacement.
• Slope of the angular velocity vs. time graph: angular acceleration.

Differences Between Linear and Rotational Variables

• Angular position is not considered a vector.
• Linear velocity and acceleration have intuitive vector directions.
• Angular velocity and acceleration vector direction is not easy to visualize; use the right-hand rule.

Relating Linear Displacement to Angular Displacement

• Linear displacement (arc length): $\Delta s$
• Angular displacement: $\Delta \theta = \frac{\Delta s}{r}$, where $r$ is the radius (length of the string).

Relating Linear Speed to Angular Speed

• Magnitude of velocity is constant at speed $v$ at points a and b, but the direction of $v$ is constantly changing.
• $v = \frac{\Delta s}{\Delta t}$
• \Delta s = \Delta \theta
  
• v = \frac{\Delta \theta
  }{\Delta t}
• $v = r \frac{\Delta \theta }{\Delta t}$
• $v = \omega r$

Linear Acceleration Components

• Centripetal acceleration: points towards the center of rotation; $a_c = \frac{v^2}{r}$
• Tangential acceleration: changes the speed of the ball rotating on the string.
• $a = \frac{\Delta v}{\Delta t}$
• $a<em>t = \frac{v</em>f - v_i}{\Delta t}$
• $a<em>t = \frac{\omega</em>f r - \omega_i r}{\Delta t}$
• $a<em>t = r \frac{\omega</em>f - \omega_i}{\Delta t}$
• $a_t = r \alpha$

Why Convert to Rotational Variables?

• Analyzing motion using angular variables simplifies the problem.
• Linear position graphs: require two functions (x and y directions) and oscillate back and forth.
• Angular displacement graphs: simplified to one dimension, resulting in a straight line.
• Linear velocity graphs: require two components (x and y directions) and oscillate back and forth.
• Angular velocity graph: a straight line in one dimension.
• Converting from linear to angular parameters reduces the workload in solving physics problems in a rotational situation.

Summary

• Relate rotational kinematic variables to translational kinematic variables.
• Converting linear to angular variables often simplifies the problem to one dimension rather than two and provides easier functions to fit the data, making problem-solving much easier.