Chapter 10 Rotation Notes

Chapter 10: Rotation

10-1 Rotational Variables

  • Learning Objectives:

    • Identify a rigid body as one where all parts rotate together around a fixed axis.
    • Define angular position as the angle of an internal reference line relative to a fixed external line.
    • Relate angular displacement to initial and final angular positions: Δθ=θ2−θ1Δθ = θ₂ - θ₁.
    • Calculate average angular velocity: ωavg=Δθ/Δtω_{avg} = Δθ/Δt.
    • Calculate average angular acceleration: αavg=Δω/Δtα_{avg} = Δω/Δt.
    • Understand that counterclockwise motion is positive and clockwise is negative.
    • Determine instantaneous angular velocity from angular position as a function of time: ω=dθ/dtω = dθ/dt.
    • Determine average angular velocity from a graph of angular position vs. time.
    • Define instantaneous angular speed as the magnitude of instantaneous angular velocity.
    • Determine instantaneous angular acceleration from angular velocity as a function of time: α=dw/dtα = dw/dt.
    • Determine average angular acceleration from a graph of angular velocity vs. time.
    • Calculate changes in angular velocity by integrating angular acceleration with respect to time.
    • Calculate changes in angular position by integrating angular velocity with respect to time.

  • Key Ideas:

    • A rigid body rotates about a fixed axis (rotation axis) with a reference line fixed in the body.
    • Angular position (θθ) is measured relative to a fixed direction.
    • When θθ is in radians: θ=s/rθ = s/r, where s is arc length and r is the radius.
    • Radian measure:
      • 1 rev=360°=2π rad1 \text{ rev} = 360° = 2π \text{ rad}.
    • Angular displacement: Δθ=θ2−θ1Δθ = θ₂ - θ₁, positive for counterclockwise, negative for clockwise rotation.
    • Average angular velocity: ωavg=Δθ/Δtω_{avg} = Δθ/Δt.
    • Instantaneous angular velocity: ω=dθ/dtω = dθ/dt.
    • Angular velocity is a vector with direction given by the right-hand rule; positive for counterclockwise rotation.
    • Angular speed is the magnitude of angular velocity.
    • Average angular acceleration: αavg=(ω2−ω1)/(t2−t1)=Δω/Δtα_{avg} = (ω₂ - ω₁)/(t₂ - t₁) = Δω/Δt.
    • Instantaneous angular acceleration: α=dw/dtα = dw/dt.
    • Both average and instantaneous angular accelerations are vectors.

What is Physics? - Rotation

  • Focus is on the motion of rotation, where objects turn about an axis.
  • Examples: machines, opening cans, amusement park rides, golf drives, curveballs, metal failure.
  • Variables for rotation are analogous to those for one-dimensional motion.
  • Important special case: constant rotational acceleration.
  • Newton's second law applies to rotational motion using torque instead of force.
  • Work and work-kinetic energy theorem apply, using rotational inertia instead of mass.
  • Many physics ideas can be applied to rotational motion with a few changes.
  • Caution:
    • Many symbols, especially Greek letters, need to be sorted out.
    • Rotation is less familiar than linear motion.
    • Translating rotational problems into linear motion problems (Chapter 2) can be helpful.

Rotational Variables Explained

  • Examines rotation of a rigid body about a fixed axis.
    • Rigid body: Parts locked together without shape change.
    • Fixed axis: Rotation axis does not move.
  • Pure rotation: every point moves in the same angle during a particular time interval.
  • Pure translation: every point moves through the same linear distance during a particular time interval.
Angular Position
  • Reference line: fixed in the body, perpendicular to the rotation axis.
  • Angular position (θθ): angle of the reference line relative to a fixed direction (zero angular position).
  • θ=s/rθ = s/r (radian measure), where s is the arc length and r is the radius (see Figures 10-2 and 10-3).
Radians
  • Angles are measured in radians (rad).
  • 1 rev=360°=2π rad1 \text{ rev} = 360° = 2π \text{ rad}
  • 1 rad=57.3°=0.159 rev1 \text{ rad} = 57.3° = 0.159 \text{ rev}
  • Angular position (θθ) is not reset to zero with each complete rotation.
  • Knowing θ(t)θ(t) completely describes the rotational motion, analogous to x(t)x(t) for linear motion.
Angular Displacement
  • Angular displacement: Δθ=θ2−θ1Δθ = θ₂ - θ₁ (change in angular position).
  • An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative. Clocks are negative.
Angular Velocity
  • Average angular velocity: ωavg=Δθ/Δtω_{avg} = Δθ/Δt.
  • Instantaneous angular velocity: ω=limΔt→0Δθ/Δt=dθ/dtω = lim_{Δt→0} Δθ/Δt = dθ/dt.
  • Units: radians per second (rad/s) or revolutions per second (rev/s) or revolutions per minute (rpm).
  • Angular velocity is positive for counterclockwise and negative for clockwise rotation (clocks are negative).
  • Angular speed is the magnitude of angular velocity.
Angular Acceleration
  • Average angular acceleration: αavg=(ω2−ω1)/(t2−t1)=Δω/Δtα_{avg} = (ω₂ - ω₁)/(t₂ - t₁) = Δω/Δt.
  • Instantaneous angular acceleration: α=limΔt→0Δω/Δt=dw/dtα = lim_{Δt→0} Δω/Δt = dw/dt.
  • Units: radians per second-squared (rad/s²) or revolutions per second-squared (rev/s²).
Sample Problem 10.01: Angular Velocity Derived from Angular Position
  • Given: θ=−1.00−0.600t+0.250t2θ = -1.00 - 0.600t + 0.250t² (t in seconds, θ in radians).
    • (a) Graph θ(t) from t = -3.0 s to t = 5.4 s; sketch reference line positions: This involves plotting the quadratic function and calculating θθ at specific times.
      • At t = -2.0 s: θ=1.2 rad=69°θ = 1.2 \text{ rad} = 69° (counterclockwise).
      • At t = 0 s: θ=−1.00 rad=−57°θ = -1.00 \text{ rad} = -57° (clockwise).
      • At t = 4.0 s: θ=0.60 rad=34°θ = 0.60 \text{ rad} = 34° (counterclockwise).
    • (b) Find the minimum θ(t). The minimum occurs at tmin where dθ/dt=0dθ/dt=0.
      • dθ/dt=−0.600+0.500t=0  ⟹  tmin=1.20 sdθ/dt = -0.600 + 0.500t = 0 \implies t_{min} = 1.20 \text{ s}.
      • θmin=−1.36 rad≈−77.9°θ_{min} = -1.36 \text{ rad} ≈ -77.9°.
    • (c) Graph angular velocity ω(t). which is the derivative and you have; sketch the disk.
      • ω=dθ/dt=−0.600+0.500tω = dθ/dt = -0.600 + 0.500t.
      • ω(−2.0 s)=−1.6 rad/sω(-2.0 \text{ s}) = -1.6 \text{ rad/s} (clockwise).
      • ω(4.0 s)=1.4 rad/sω(4.0 \text{ s}) = 1.4 \text{ rad/s} (counterclockwise).
      • At tmint_{min}, ω=0ω = 0.
    • (d) Describe the motion: The disk initially rotates clockwise while slowing down, then stops and rotates counterclockwise.
    • Disk has positive angular position
    • Turning clockwise and slowing, eventually becomes positive again
Sample Problem 10.02: Angular Velocity Derived from Angular Acceleration
  • Given: α=5t3−4tα = 5t³ - 4t, ω(0)=5 rad/sω(0) = 5 \text{ rad/s}, θ(0)=2 radθ(0) = 2 \text{ rad}.
    • (a) Find ω(t) by integrating α(t):
      • ω=∫αdt=∫(5t3−4t)dt=54t4−2t2+Cω = ∫α dt = ∫(5t³ - 4t) dt = \frac{5}{4}t^4 - 2t² + C.
      • Using ω(0)=5 rad/sω(0) = 5 \text{ rad/s}, C=5C = 5, so ω(t)=54t4−2t2+5ω(\text{t}) = \frac{5}{4}t^4 - 2t² + 5.
    • (b) Obtain an expression for angular position θ(t).
      • θ=∫ωdt=∫(54t4−2t2+5)dt=t54−23t3+5t+C′θ = ∫ω dt = ∫(\frac{5}{4}t^4 - 2t² + 5) dt = \frac{t^5}{4} - \frac{2}{3}t^3 + 5t + C'
      • Given θ(0)=2θ(0)=2, C′=2C' = 2. Therefore, θ(t)=t54−23t3+5t+2θ(t) = \frac{t^5}{4} - \frac{2}{3}t^3 + 5t + 2
Vector Properties of Angular Quantities
  • Similar to linear motion, angular quantities can be treated as vectors, but with caveats.
Angular Velocities
  • Angular velocity (ωω) can be represented as a vector along the axis of rotation.
  • Right-hand rule: Curl fingers in the rotation direction, thumb points along the ωω vector.
  • ωω vector defines the rotation axis but not a direction of movement along the axis.
  • The angular quantities obey all vector manipulation rules discussed in Chapter 3
Angular Displacements
  • Angular displacements (unless very small) cannot be treated as vectors because vector addition is not commutative.
  • The order in which angular displacements are added matters.

10-2 Rotation with Constant Angular Acceleration

  • If angular acceleration is constant, we have:
    • ω=ω0+αtω = ω₀ + αt
    • θ−θ0=ω0t+12αt2θ - θ₀ = ω₀t + \frac{1}{2}αt²
    • ω2=ω02+2α(θ−θ0)ω² = ω₀² + 2α(θ - θ₀)
    • θ−θ0=12(ω0+ω)tθ - θ₀ = \frac{1}{2}(ω₀ + ω)t
    • θ−θ0=ωt−12αt2θ - θ₀ = ωt - \frac{1}{2}αt²
Sample Problem 10.03: Constant Angular Acceleration, Grindstone
  • Given: α=0.35 rad/s²α = 0.35 \text{ rad/s²}, ω0=−4.6 rad/sω₀ = -4.6 \text{ rad/s}, θ0=0θ₀ = 0.
    • (a) At what time is θ=5.0 revθ = 5.0 \text{ rev}?
      • 5.0 rev=10π rad5.0 \text{ rev} = 10π \text{ rad}.
      • Using θ−θ0=ω0t+12αt2θ - θ₀ = ω₀t + \frac{1}{2}αt²:
      • 10π=−4.6t+12(0.35)t2  ⟹  t=32 s10π = -4.6t + \frac{1}{2}(0.35)t² \implies t = 32 \text{ s}.
    • (b) Describe the rotation:
      • The wheel initially slows down, reverses direction, and then rotates in the positive direction.
    • (c) When does the grindstone stop momentarily?
      • Using ω=ω0+αt  ⟹  0=−4.6+0.35t  ⟹  t=13 sω = ω₀ + αt \implies 0 = -4.6 + 0.35t \implies t = 13 \text{ s}.
Sample Problem 10.04: Constant Angular Acceleration, Riding a Rotor
  • ω0=3.40 rad/sω₀ = 3.40 \text{ rad/s}, ω=2.00 rad/sω = 2.00 \text{ rad/s}, Δθ=20.0 revΔθ = 20.0 \text{ rev}.
    • (a) Constant angular acceleration α: * t = \frac{ω - ω₀}{α}
      • Substitute into :θ−θ0=ω0t+12αt2θ - θ₀ = ω₀t + \frac{1}{2}αt²\quad then:   ⟹  α=(ω2−ω02)/(2Δθ)=−0.0301 rad/s²\implies α = (ω² - ω₀²)/(2Δθ) = -0.0301 \text{ rad/s²}.
    • (b) Time to decrease speed: