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: extRadians=extDegreesimesπ180ext{Radians} = ext{Degrees} imes \frac{\pi}{180}

    • Radians to Degrees: extDegrees=extRadians×180πext{Degrees} = ext{Radians} \times \frac{180}{\pi}

  • Angular Velocity (ω\omega):

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

  • Angular Acceleration (α\alpha):

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

Linear vs. Angular Equations

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

    • s=R×θs = R \times \theta

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

    • vt=ω×Rv_t = \omega \times R

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

    • at=α×Ra_t = \alpha \times R

Acceleration Types

  • Tangential Acceleration ($a_t$):

    • Change in velocity.

  • Radial Acceleration ($a_r$):

    • a<em>r=v</em>t2Ra<em>r = \frac{v</em>t^2}{R}.

    • Always points toward the center of the circle.

Torque and Dynamics

  • Torque (τ\tau):

    • τ=r×F×sin⁡(θ)\tau = r \times F \times \sin(\theta) (where θ\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:

    • Work=τ×θ\text{Work} = \tau \times \theta.

  • Power in Rotation:

    • P=τ×ωP = \tau \times \omega.

  • Rotational Kinetic Energy ($KE$):

    • KE=12Iω2KE = \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=12mR2I = \frac{1}{2} m R^2.

  • Conservation of Angular Momentum:

    • I<em>1ω</em>1=I<em>2ω</em>2I<em>1 \omega</em>1 = I<em>2 \omega</em>2 (before and after interactions).

Harmonic Motion Relations

  • Frequency ($f$) and Period ($T$):

    • f=1Tf = \frac{1}{T} and ω=2πf\omega = 2\pi f.

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

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