Dynamics of a Hand-Held Ball-on-a-String — Interactive 3D Physics Lab

Experiment Presets

Select a physical boundary condition

System Parameters

Manipulate Sections 1 & 3 variables

Initial Velocity (ω_i) 120 RPM (12.6 rad/s)

5 RPMDefault: 120 RPM1200 RPM

Initial String Radius (r_i) 0.50 m

0.05 mDefault: 0.50 m3.00 m

Final String Radius (r_f) 0.050 m (10.0×)

0.005 mDefault: 0.05 m1.50 m

Pull Duration (Δt) 1.00 s (|dr/dt|=0.45 m/s)

0.05 sDefault: 1.00 s30.00 s

Ball Mass (m) 0.050 kg (50 g)

1 gDefault: 50 g1000 g (1 kg)

Ball Diameter (d) 0.040 m (4.0 cm)

0.2 cmDefault: 4.0 cm30.0 cm

Table 1 Comparison @ r = r_f dL/dt = τ_net

Metric	Ideal	Bound	Sim Peak
Peak ω_f	12,000	427 RPM	378 RPM
Tension F_c	3,948 N	5.0 N	3.9 N
Wobble R_eq	0.0 mm	12.0 mm	11.6 mm
L_sys Retained	100%	7.6%	6.5%
L_ball Retained	100%	3.56%	3.14%

Speed:

t = 0.00 s Pull: 0→1.0s | Stop: 4.2s

PHASE 1: INWARD PULL (0 ≤ t ≤ Δt)

Ball & Trail Pinch Wobble R(r,ω) at y=0 Centripetal F_c & Wrist Heat Tangential v⃗

Drag: Orbit 3D | Right-Drag / Shift-Drag: Pan | Scroll: Zoom

Real-Time Trajectory & Post-Pull Coast-Down Plots (Click or Drag Any Plot to Scrub Time) Shaded Cyan = Inward Pull (0 ≤ t ≤ Δt) | Right of Dashed Line = Post-Pull Decay & Stop (t > Δt)

1. Angular Velocity ω(t) [RPM] 120 RPM

2. Wobble R(t) [mm] & String r(t) [m] R=10.8mm, r=0.50m

3. Torque Balance τ_spin-up vs |τ_net| [N·m] τ_net = -0.04 N·m

Angular Velocity | Radii | String Tension

Live telemetry

t = 0.00 s

Angular Velocity

120.0 RPM

12.57 rad/s

Ideal: 120 RPM

Radii (r & R)

0.500 m

Wobble R: 10.8 mm

Yield ΔR: +0.8 mm

String Tension F_c

3.95 N

Droop φ: 7.1°

Ideal: 3.95 N

Angular Momentum Partition (L = Iω)

Coupled two-mass inertial sink & external bleed

Initial System L_sys,i: 0.1583 N·m·s

Current System Retained: 0.1583 N·m·s (100.0%)

Ball Share L_ball 0.1571 N·m·s (99.2%)

Hand Wobble Share L_hand (M R² ω) 0.0013 N·m·s (0.8%)

Bled Off by External Torques (∫τ_netdt) 0.0000 N·m·s (0.0%)

Instantaneous Torque Balance (N·m)

I_eff (dω/dt) = τ_spin-up + τ_net

+ Coriolis Inward Spin-Up (τ_spin-up) +0.2827 N·m

− Wrist Viscous Damping (τ_damp ∝ ω⁵) -0.0229 N·m

− Finger Pinch Skin Friction (τ_fric) -0.0063 N·m

− Aero Drag Ball + String (τ_drag) -0.0102 N·m

Net Resistive Torque (τ_net) -0.0394 N·m

Wrist Heat Dissipation (P_diss = |τ_damp|ω) 0.29 W