Problem 8 IYPT 2027

8. Upward-driven disc

Mechanics Oscillations

Problem Description & Analysis

Official Problem Statement

A disc is placed between two spinning wheels that drive it upward. The disc can oscillate about the contact point. Investigate the phenomenon.

Apparatus Geometry

The disc is vertical — like a coin standing on its edge. Two cylindrical rollers grip the flat faces of the disc from either side; the disc rim is not in the groove between the rollers.

Forces each roller applies to the disc face:
• Normal N: horizontal, clamps the disc between the two rollers
• Upward friction f = μN: the roller surface moves upward (via rotation), dragging the disc face upward
• Contact point O: where the roller meets the disc face; if O is above the disc’s center of mass, the disc behaves as a simple pendulum

Core Physics — Mechanical Pendulum

Gyroscopic effects play no dominant role here. The rollers apply upward friction to the disc face — they do not impart significant spin to the disc about its own axis, so the disc’s angular momentum is negligible. The dominant mechanism is a simple pendulum:

Let h be the vertical distance from the contact point O down to the disc’s center of mass CM. When the disc tilts by angle θ, gravity provides a restoring torque — exactly like a pendulum hanging from O:

$$I_O,ddot{theta} = -Mg,h,sintheta$$

Moment of inertia about O (parallel-axis theorem):

$$I_O = underbrace{tfrac{1}{4}MR^2}_{I_{CM}} + M h^2$$

Natural oscillation frequency for small angles:

$$boxed{omega_0 = sqrt{frac{Mgh}{I_O}} = sqrt{frac{gh}{tfrac{1}{4}R^2 + h^2}}}$$

System Parameters

• M — disc mass (kg)
• R — disc radius (m)
• h — vertical distance from contact point O to CM (m) — the key parameter
• I_O = ¼MR² + Mh² — moment of inertia about tilt axis O
• N — normal (clamping) force from each roller (N)
• μ — roller surface friction coefficient
• f = μN — upward friction force from each roller

Vertical (Rising) Motion

The disc rises when the combined upward friction exceeds its weight:

$$2mu N > Mg quadRightarrowquad N > frac{Mg}{2mu}$$

Upward acceleration when this condition holds:

$$a = frac{2mu N – Mg}{M}$$

The vertical (rising) and transverse (tilting) motions are largely decoupled for small oscillations. Their coupling at certain driving speeds produces the chaotic behavior reported in AJP 2025.

Three Behavioral Regimes

Regime Condition Disc Behavior
Steady rise θ ≈ 0, perfect symmetry Disc rises without tilting
Pendulum oscillation h > 0, O above CM Disc swings about O while rising steadily
Unstable h ≤ 0, O at or below CM Disc falls (inverted pendulum — no restoring force)
Chaotic Specific driving speeds No periodic pattern (AJP 2025)

Numerical Example

R = 6 cm, h = 3 cm (contact point half a radius above CM):

$I_O = tfrac{1}{4}(M)(0.06)^2 + M(0.03)^2 = 1.8times10^{-3}M$ kg·m²

$omega_0 = sqrt{9.81 times 0.03 / (1.8times10^{-3})} approx 12.8$ rad/s ≈ 122 RPM

Oscillation period: $T = 2pi/omega_0 approx 0.49$ s

Experimental Setup

Construction

  • Rollers: Two cylinders with parallel horizontal axes; rubber or velvet surface for high friction; r_roller = 2–4 cm.
  • Drive: Two DC motors with PWM controller, both rotating so that their contact surfaces move upward.
  • Disc: Aluminium or steel, vertical; R = 5–10 cm; m = 50–200 g; must be rotationally balanced.
  • Adjustable height: The roller contact height relative to disc center (parameter h) must be adjustable — this is the most important experimental variable.
  • Safety guard: The disc may be launched sideways — install a transparent guard around the apparatus.

Measurement

  • 120fps camera: Track tilt angle θ(t) and oscillation frequency via video analysis.
  • IMU (MPU-6050): Gyro + accelerometer mounted at disc center — records θ(t) directly.
  • Experimental design: Vary h (roller contact height) and measure oscillation frequency. Compare with pendulum model prediction ω₀ = √(gh / (R²/4 + h²)).
️ The disc may be launched outward — install a transparent plastic guard.
️ Roller axes must be parallel to within 0.5 mm.
️ Both rollers must be at the same height to maintain symmetric contact.

Deeper Analysis — Why Friction Does Not Affect the Oscillation Frequency

A subtle but important point: the upward friction from the rollers has zero effect on the oscillation frequency. The reasoning: we write the torque equation about the contact point O. The friction force f = μN is applied at O, so its moment arm about O is exactly zero:

$$tau_f^{(O)} = vec{r}_{Oto O} times vec{f} = 0$$

Only gravity contributes a restoring torque, with moment arm $hsintheta$. Therefore:

Key result: the oscillation frequency $omega_0 = sqrt{gh/(R^2/4+h^2)}$ is completely independent of the clamping force N and friction coefficient μ. Changing the roller speed does not alter the pendulum frequency — only the contact height h matters.

Optimal Contact Height — Maximum Oscillation Frequency

Setting $d(omega_0^2)/dh = 0$:

$$frac{d}{dh}!left[frac{gh}{tfrac{R^2}{4}+h^2}right] = g,frac{tfrac{R^2}{4}-h^2}{left(tfrac{R^2}{4}+h^2right)^2} = 0 quadRightarrowquad boxed{h_text{opt} = frac{R}{2}}$$

At $h = R/2$ the oscillation frequency is maximum:

$$omega_{0,max} = sqrt{frac{g cdot R/2}{tfrac{R^2}{4}+tfrac{R^2}{4}}} = sqrt{frac{g}{R}}$$

Physical interpretation: $omega_{0,max} = sqrt{g/R}$ equals the frequency of a simple pendulum of length $R$. At $h = R/2$ the gravitational torque and moment of inertia balance to give maximum frequency.

• $h to 0$: O approaches CM → zero restoring torque → zero frequency (disc falls without oscillating).
• $h to R$: large moment of inertia → frequency falls again.

Interactive Python Simulation

Three panels generated by numerically solving the disc pendulum ODE:

  • Panel 1: $omega_0/omega_text{max}$ vs $h/R$ — bell-shaped curve peaking at $h = R/2$
  • Panel 2: $theta(t)$ for five contact heights — clearly different periods
  • Panel 3: Phase portrait $(theta, dottheta)$ — closed orbits for stable oscillation
Key results from the simulation:

  • Table: maximum frequency at h = 0.5R; drops sharply for h = 0.1R or h = R.
  • Panel 2: same initial amplitude (18°), clearly different periods — confirms the pendulum model prediction.
  • Panel 3: phase ellipses are tightest for h = 0.5R (highest frequency) and widest for extreme h values.

Video Resources

References

  1. Hartl, D.J., & Lutsko, J.F. (2025). The upward-driven disk: a steadily forced chaotic system. American Journal of Physics, 93(8), 620–630.
  2. Goldstein, H., Poole, C., & Safko, J. (2002). Classical Mechanics, 3rd ed. Addison-Wesley.
  3. Landau, L.D., & Lifshitz, E.M. (1982). Mechanics, 3rd ed. Pergamon Press.
  4. IYPT 2027 Problem 8 Official Statement. iypt.org

Technical Glossary

Term Persian equivalent Definition
Gyroscopic effect اثر ژیروسکوپی Tendency of a fast-spinning body to maintain its rotation axis orientation; resists precession
Precession پرسسیون Slow rotation of the spin axis under a perpendicular torque: $Omega_p = tau / L$
Angular momentum تکانه‌ی زاویه‌ای $mathbf{L} = Iboldsymbol{omega}$; conserved when net torque = 0; keeps the disc axis stable
Torque گشتاور Rotational analogue of force: $boldsymbol{tau} = mathbf{r} times mathbf{F}$; equals the rate of change of angular momentum
Rolling contact (point O) تماس غلشی (نقطه‌ی O) The contact point with the surface; friction force at O produces zero torque about O → frequency independent of μ
Moment of inertia ممان اینرسی Resistance to angular acceleration; solid disc about centre: $I = tfrac{1}{2}MR^2$
Optimal height $h$ ارتفاع بهینه $h$ Distance from CoM to contact point O; angular frequency is maximised at $h = R/2$

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