Rotational Motion Study Notes

Rotational Motion Overview

• Circle Basics:

  • Radius: Distance from center to any point on circle.

  • Arc Length ($s$): Distance between two points on circle; calculated as $s = heta imes R$ (where $ heta$ is in radians).

  • 1 radian: Angle subtended by arc length equal to the radius.

Angular Concepts

• Angle Conversion:

  • Degrees to Radians: $ ext{Radians} = ext{Degrees} imes \frac{\pi}{180}$

  • Radians to Degrees: $ ext{Degrees} = ext{Radians} \times \frac{180}{\pi}$

• Angular Velocity ($\omega$):

  • Defined as $\omega = \frac{\Delta \theta}{\Delta t}$ with units in radians/second.

• Angular Acceleration ($\alpha$):

  • Defined as $\alpha = \frac{\Delta \omega}{\Delta t}$.

Linear vs. Angular Equations

• Linear Displacement ($s$) and Angular Displacement ($\theta$):

  • $s = R \times \theta$

• Tangential Velocity ($v_t$) and Angular Velocity ($\omega$):

  • $v_t = \omega \times R$

• Tangential Acceleration ($a_t$) and Angular Acceleration ($\alpha$):

  • $a_t = \alpha \times R$

Acceleration Types

• Tangential Acceleration ($a_t$):

  • Change in velocity.

• Radial Acceleration ($a_r$):

  • $ar = \frac{vt^2}{R}$.

  • Always points toward the center of the circle.

Torque and Dynamics

• Torque ($\tau$):

  • $\tau = r \times F \times \sin(\theta)$ (where $\theta$ is the angle between $F$ and the radius).

  • Based on which way the force is applied on an object.

• Net Torque:

  • Analyzing multiple forces to determine the overall rotational effect.

Energy in Rotational Motion

• Work Done by Torque:

  • $\text{Work} = \tau \times \theta$.

• Power in Rotation:

  • $P = \tau \times \omega$.

• Rotational Kinetic Energy ($KE$):

  • $KE = \frac{1}{2} I \omega^2$ where $I$ is the moment of inertia.

Frequently Used Equations

• Moment of Inertia ($I$):

  • For a solid disk: $I = \frac{1}{2} m R^2$.

• Conservation of Angular Momentum:

  • $I1 \omega1 = I2 \omega2$ (before and after interactions).

Harmonic Motion Relations

• Frequency ($f$) and Period ($T$):

  • $f = \frac{1}{T}$ and $\omega = 2\pi f$.

• Angular Displacement ($\theta$) in Harmonic Motion:

  • $\theta = \omega t$ (with respect to time).