Year-13 Physics — Rotational Dynamics & Banked Curves

Banked-Curve & Circular-Motion Applications

  • Design question 5

    • Required banking angle for a roadway when
    • Design speed v = 13.2\,\text{m\,s}^{-1}
    • Curve radius r = 48.0\,\text{m}
    • Friction-free condition implies
      \tan\theta = \frac{v^{2}}{rg}
    • Used in road safety, racetrack design, aviation (tilted wings).

  • Design question 6 (race-track)

    • Two limiting radii on a concrete, friction-free, banked oval
    • Inner lane r_{\min}=112\,\text{m}
    • Outer lane r_{\max}=165\,\text{m}
    • Wall height h = 18\,\text{m} allows determination of banking angle by simple trigonometry (rise/run). Once \theta is fixed the usable speed range is
      v{\min}=\sqrt{rg\tan\theta}\,,\qquad v{\max}=\sqrt{rg\tan\theta}\,\Big|{r=r{\max}}
    • Shows how a single surface can be safe for different radii.

  • Design question 7 (Ferris wheel)

    • Child mass m=40\,\text{kg}, radius r=10\,\text{m}, constant speed v=3\,\text{m\,s}^{-1}.
    • Forces on rider
    • Bottom: N_{\text{bottom}}=mg+\frac{mv^{2}}{r} (heavier feeling)
    • Top: N_{\text{top}} = mg-\frac{mv^{2}}{r} (lighter feeling).
    • Demonstrates interplay of apparent weight and centripetal requirement.

Rotational Dynamics — Motivation

  • Extended bodies (wheels, planets, motors) cannot be analysed as point masses because their particles have differing linear speeds.
  • Need rotational analogues of displacement, velocity, acceleration, force.

Angular Variables

Angular Displacement \theta

  • Measured in radians: \theta = \frac{s}{r} (arc length over radius).
  • One revolution =2\pi\,\text{rad}.

Angular Velocity \omega

  • Rate of change of angular displacement: \omega = \frac{d\theta}{dt} (units \text{rad\,s}^{-1}).
  • Linear connection: v = r\omega.
  • Frequency relation: \omega = 2\pi f.

Angular Acceleration \alpha

  • Rate of change of angular velocity: \alpha=\frac{d\omega}{dt} (units \text{rad\,s}^{-2}).
  • If \omega changes uniformly: \alpha = \frac{\omega{f}-\omega{i}}{\Delta t}.
  • Linear link: a_{t}=r\alpha (tangential acceleration).

Uniform Angular Acceleration — Kinematic Equations

  • Exact analogues of linear s,u,v,a,t equations.
    • \omega{f}=\omega{i}+\alpha t
    • \theta = \omega_{i}t + \tfrac12 \alpha t^{2}
    • \omega{f}^{2}=\omega{i}^{2}+2\alpha\theta

Linear↔Angular Variable Table

  • Speed: u,v\;\leftrightarrow\;\omega{i},\omega{f}
  • Displacement: s\;\leftrightarrow\;\theta
  • Acceleration: a\;\leftrightarrow\;\alpha

Worked Example 1.13 — Photographic Record

  • Starts from rest, reaches 45\,\text{rev\,min}^{-1} (i.e. 4.71\,\text{rad\,s}^{-1}) in 5\,\text{s}.
    • Angular acceleration \alpha=0.94\,\text{rad\,s}^{-2}.
    • Revolutions covered in this startup =11.75\,\text{rev}.

Practice Problems (Exercise 1.5)

  • Convert Earth’s orbital period T=3.156\times10^{7}\,\text{s} to \omega_{\text{earth}}.
  • Wheel speed on a moving bicycle, v=7.5\,\text{m\,s}^{-1}, r=0.30\,\text{m}.
  • Flywheel deceleration from 500\,\text{Hz} to rest in 5\,\text{s}.
  • Car wheel slowdown: r=0.36\,\text{m}, v=18\,\text{m\,s}^{-1}, N=25\,\text{turns}.
  • etc.

Moment of Inertia I

  • Rotational analogue of mass: I=\sum m{i} r{i}^{2}.
  • Depends on mass distribution relative to axis.
  • Units \text{kg\,m}^{2}.

Standard Shapes (about stated axes)

  • Hoop / thin cylindrical shell: I = MR^{2}.
  • Solid cylinder/disc: I = \tfrac12 MR^{2}.
  • Hollow cylinder: I = \tfrac12 M\left(R{1}^{2}+R{2}^{2}\right).
  • Thin rod (centre): I = \tfrac1{12} ML^{2}.
  • Thin rod (end): I = \tfrac13 ML^{2}.
  • Solid sphere: I = \tfrac25 MR^{2}.
  • Thin spherical shell: I = \tfrac23 MR^{2}.
  • Rectangular plate (axis through centre, perpendicular): I = \tfrac1{12}M(a^{2}+b^{2}).

Examples

  1. Point mass m=2\,\text{kg} on string r=0.6\,\text{m} ⇒ I=0.72\,\text{kg\,m}^{2}.
  2. Two-mass dumb-bell: m{1}=2\,\text{kg}, m{2}=3\,\text{kg}, L=1\,\text{m}. About midpoint ⟂ to rod:
    I = m{1}\left(\tfrac12L\right)^{2}+m{2}\left(\tfrac12L\right)^{2}=1.25\,\text{kg\,m}^{2}.

Torque \tau

  • Rotational effectiveness of a force: \tau = rF_{\perp} = I\alpha (units \text{N\,m}).
  • Produced by string tension in falling-mass apparatus, figure-of-eight belts, engine crankshafts.

Torque–Acceleration Derivation (Example 1.15)

  • Uniform disc I=\tfrac12 m r^{2} hanging by wrapped string.
  • Translational mg - T = ma; rotational \tau = Tr = I\alpha with a=r\alpha.
  • Solve ⇒ linear acceleration a = \tfrac23 g(i.e., two-thirds of free fall).
  • Demonstrates energy sharing: some gravitational PE goes to rotational KE.

Angular Momentum L

  • For rigid body: L = I\omega.
  • For point mass: L = r m v = m r^{2} \omega (direction via right-hand rule).
  • Units \text{kg\,m}^{2}\,\text{s}^{-1}.

Conservation Principle

  • If external torque \sum \tau{\text{ext}} = 0, then L{\text{initial}} = L_{\text{final}}.
  • Enables ice-skater spin-ups, neutron-star dynamo predictions, merging-disk problems.

Worked Examples (1.16)

  1. Wheel accelerated by \tau=100\,\text{N\,m} from rest to \omega=20\,\text{rad\,s}^{-1} over \theta=10\,\text{rad}.
    • Using \omega^{2}=2\alpha\theta ⇒ \alpha=20\,\text{rad\,s}^{-2}.
    • Moment of inertia from \tau=I\alpha ⇒ I=5\,\text{kg\,m}^{2}.
    • Angular momentum gained \Delta L = I\omega = 100\,\text{kg\,m}^{2}\text{s}^{-1}.
  2. Two coaxial disks couple (clutch problem).
    • Disk A: I{1}=2\,\text{kg\,m}^{2}, \omega{1}=10\,\text{rad\,s}^{-1}.
    • Disk B: I{2}=3\,\text{kg\,m}^{2}, \omega{2}=0.
    • After drop: \omega{f}=\tfrac{I{1}\omega{1}+I{2}\omega{2}}{I{1}+I_{2}}=4\,\text{rad\,s}^{-1}.

Additional Exercises (1.6)

  • Turntable–record coupling, I{1}=0.09, I{2}=0.03\,\text{kg\,m}^{2}.
  • Cylindrical spacecraft attitude control via expelled gas: each puff m=0.4\,\text{kg}, v=100\,\text{m\,s}^{-1} at r=2\,\text{m}.
    • Linear momentum per puff p=mv.
    • Angular momentum delivered L=rp, conserve to find craft’s \Delta\omega.

Conceptual & Real-World Connections

  • Banking removes dependence on tyre friction ⇒ higher safety margins.
  • Ferris-wheel apparent-weight changes underpin design of safe lap-bars.
  • Disk-clutch coupling is mechanical analogue of perfectly inelastic collision.
  • Moment of inertia critical in flywheels (energy storage) & figure-skating (artistic spins).
  • Torque link to power P=\tau\omega governs engine ratings.
  • Conservation of angular momentum fundamental in astrophysics (planet formation, pulsars).

Ethical & Practical Notes

  • Proper banking saves lives by reducing skidding accidents.
  • Amusement-ride designers must account for human tolerance to varying normal forces.
  • Rocket attitude thrusters must conserve propellant while delivering required angular impulse.