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
    $v<em>{\min}=\sqrt{rg\tan\theta}\,,\qquad v</em>{\max}=\sqrt{rg\tan\theta}\,\Big|<em>{r=r</em>{\max}}$
  • 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: $\alpha = \frac{\omega<em>{f}-\omega</em>{i}}{\Delta t}$.
• Linear link: $a_{t}=r\alpha$ (tangential acceleration).

Uniform Angular Acceleration — Kinematic Equations

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

Linear↔Angular Variable Table

• Speed: $u,v\;\leftrightarrow\;\omega<em>{i},\omega</em>{f}$
• 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: $I=\sum m<em>{i} r</em>{i}^{2}$.
• 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: $I = \tfrac12 M\left(R<em>{1}^{2}+R</em>{2}^{2}\right)$.
• 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: $m<em>{1}=2\,\text{kg}, m</em>{2}=3\,\text{kg}, L=1\,\text{m}$. About midpoint ⟂ to rod:
   $I = m<em>{1}\left(\tfrac12L\right)^{2}+m</em>{2}\left(\tfrac12L\right)^{2}=1.25\,\text{kg\,m}^{2}$.

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 $\sum \tau<em>{\text{ext}} = 0$, then $L</em>{\text{initial}} = L_{\text{final}}$.
• 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: $I<em>{1}=2\,\text{kg\,m}^{2}, \omega</em>{1}=10\,\text{rad\,s}^{-1}$.
   • Disk B: $I<em>{2}=3\,\text{kg\,m}^{2}, \omega</em>{2}=0$.
   • After drop: $\omega<em>{f}=\tfrac{I</em>{1}\omega<em>{1}+I</em>{2}\omega<em>{2}}{I</em>{1}+I_{2}}=4\,\text{rad\,s}^{-1}$.

Additional Exercises (1.6)

• Turntable–record coupling, $I<em>{1}=0.09, I</em>{2}=0.03\,\text{kg\,m}^{2}$.
• 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.