Rotational Kinematics Notes

Rotational Motion and Angular Displacement

  • Definition: When a rigid body rotates around a fixed axis, each point moves on a circular path.

  • Angular Displacement (Δθ): The angle through which the object rotates.

    • Positive if counterclockwise

    • Negative if clockwise

    • SI Unit: Radian (rad)

    • 1 complete revolution = 2π2\pi radians

    • Degrees relationship: 360°=2πextrad360° = 2\pi ext{ rad}

Arc Length and Angular Displacement

  • Arc Length (s): The linear distance traveled by a point on the rotating object related to its radius (r) and angular displacement (θ).

    • Formula: s=rθs = r \theta

  • Full Revolution:

    • s=2πrs = 2\pi r

Units of Angular Displacement

  1. Degrees: Full circle = 360°

  2. Revolution (rev): One complete turn of 360°

  3. Radian (rad): An angle where the arc length equals the radius.

Example: Synchronous Satellites

  • Two satellites in orbit with radius r=7.423×107mr = 7.423 \times 10^7 m and angular separation of 2.00°.

  • Arc Length Calculation:

    • Convert degrees to radians:
      θ=2.00°×π180°=0.0349extrad\theta = 2.00° \times \frac{\pi}{180°} = 0.0349 ext{ rad}

    • Arc Length (s):

    • s=rθ=(7.423×107m)(0.0349extrad)=1.48×106ms = r\theta = (7.423 \times 10^7 m)(0.0349 ext{ rad}) = 1.48 \times 10^6 m

Angular Velocity and Angular Acceleration

Definition of Average Angular Velocity

  • Formula: ω=ΔθΔt\omega = \frac{\Delta \theta}{\Delta t}

  • SI Unit: radian per second (rad/s)

Example: Gymnast on a High Bar

  • A gymnast completes 2 revolutions in 1.90 s.

  • Average Angular Velocity Calculation:

    • Total angular displacement for 2 revs: 2×2πextrad=12.6extrad2 \times 2\pi ext{ rad} = 12.6 ext{ rad}

    • ω=12.6extrad1.90exts=6.63extrad/s\omega = \frac{12.6 ext{ rad}}{1.90 ext{ s}} = 6.63 ext{ rad/s}

Instantaneous Angular Velocity

  • Definition: The limit of average angular velocity as the time interval approaches zero.

  • Notation: ω=lim⁡Δt→0ΔθΔt\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}

Angular Acceleration

  • Definition of Average Angular Acceleration:

  • Formula: a=ΔωΔta = \frac{\Delta \omega}{\Delta t}

  • SI Unit: radian per second squared (rad/s²)

Example: Jet Engine

  • Angular velocity changes from −110extrad/s-110 ext{ rad/s} to −330extrad/s-330 ext{ rad/s} in 14 s.

  • Angular Acceleration Calculation:
    a=−330−(−110)14=−16extrad/s2a = \frac{-330 - (-110)}{14} = -16 ext{ rad/s²}

Equations of Rotational Kinematics

Kinematic Variables for Rotational Motion:

  1. Displacement: θ\theta

  2. Initial Angular Velocity: ω0\omega_0

  3. Final Angular Velocity: ω\omega

  4. Angular Acceleration: α\alpha

  5. Time: tt

Key Equations of Rotational Kinematics (for constant angular acceleration):

  1. θ=ω0t+12αt2\theta = \omega_0 t + \frac{1}{2} \alpha t^2

  2. ω=ω0+αt\omega = \omega_0 + \alpha t

  3. ω2=ω02+2αθ\omega^2 = \omega_0^2 + 2\alpha\theta

Reasoning Strategy for Problem Solving

  1. Draw a diagram.

  2. Define positive and negative directions (CCW = positive).

  3. List known values for kinematic variables.

  4. Ensure at least three variables are known to select the appropriate equation.

  5. Remember that the final angular velocity of one segment becomes the initial angular velocity of the next.

  6. Two possible answers may exist for the kinematics problem.

Example Problem: Blender

  • Angular velocity: +375 rad/s.

  • Angular acceleration: 1740 rad/s² for an angular displacement of +44.0 rad.

  • Final Angular Velocity Calculation:

    • Use: ω2=ω02+2αθ\omega^2 = \omega_0^2 + 2\alpha\theta

    • Final result yields ω=542extrad/s\omega = 542 ext{ rad/s}

These notes encompass the concepts, definitions, and examples found within "Chapter 8: Rotational Kinematics" from Cutnell & Johnson's Physics 12th Edition.