Hooked Fish Lab: Rod Bending, Line Stretch and Drag Clutch

A rod clamped on a pier, an elastic line, a slipping reel clutch and a fish that pulls, tires and displaces water. Find the drag setting that lands the fish without snapping the line.

Presets

Hold Reel (or key Space) to wind in; ↑/↓ raise and lower the rod to pump. If tension passes the clutch setting the spool slips and line runs out; past the line's breaking load it snaps. Drag to orbit, wheel / pinch to zoom.

Rod

Line and reel

Fish

World

Display

Before the bite: a spherical float carries a lead sinker. The float sits so that the weight of displaced water equals the net weight it carries (Archimedes). A fish tug that exceeds the reserve buoyancy pulls the float under, which is the visible bite.

Rig

Readout

Float draught over time (mm)

draughtArchimedes equilibrium

Line tension and its limits (N)

tensionclutchbreaking load

Fish distance and line paid out (m)

fish to rod tipline paid out by the clutch (total)

Fight ledger

1 - How far does a loaded rod bend?

A horizontal cantilever rod, tip load F. Simulated tip deflection (large-deflection elastica) against the beam formula δ = F L³ / (3 EI) for a uniform rod and the unit-load integral for the tapered rod. Small bends agree; large bends fall short of the linear formula because the rod shortens as it curves.

2 - Archimedes: where does the float settle, and how fast does a sinker fall?

The float is released from the surface and left to settle in fresh, sea and Dead Sea water; its draught is compared with the solution of ρ_w V_cap(h) = m_float + m_sinker(1 - ρ_w/ρ_lead). A free sinker's terminal speed is compared with √(2 W' / (ρ_w C_d A)).

3 - Which clutch setting lands the fish?

Same fish, rod, line and (auto) angler for each run; only the clutch setting changes, from loose to tight. Look for the window between "fish runs away with all the line" and "line snaps". Each run is simulated up to 100 s, so this takes a few seconds.

The science

Rod as a cantilever. A tapered tube clamped at the butt: bending stiffness EI(s) = EI₀ (1 - (1-r) s/L)⁴ (wall thickness fixed, diameter shrinking linearly, so I ∝ d⁴). The curvature is dθ/ds = M(s)/EI(s) with M the moment of the tip force about the point s. For a uniform rod and small bends this gives δ = F L³ / 3EI; the simulation solves the full nonlinear (elastica) problem by fixed-point iteration, so a hard-pulled rod curves and shortens.

Line as a spring. Tension T = EA ε + c ε' when the line is taut (ε = (distance - paid-out length)/paid-out length), zero when slack. The breaking load fixes EA through the strain at break (nylon about 25%, braid about 4%), so a long mono line is a soft spring and absorbs shock; braid transmits spikes. Longer line = lower stiffness EA/L.

Drag clutch. When T exceeds the clutch setting T_c the spool slips and pays out line at (T - T_c)/R (R is the slip compliance). Tension therefore stays near T_c while the fish runs. Winding in works only below T_c and is limited by the angler's cranking force. Anglers set the clutch around 25-35% of the line's breaking load.

Fish. Mass m, length 0.4 m^1/3 m. Thrust is bursty: F = k m g u, with u falling as stamina is spent and recovering slowly at rest. Drag is anisotropic, ½ ρ_w (C_fA_fcos²α + C_sA_ssin²α) v², slim along the body and broad across it, which is why a fish turned sideways tires fast. Terminal speed against thrust F: v = √(2F/ρCA). Virtual mass 1.15 m.

Archimedes. Net weight in water (m - ρ_w V) g; a fish denser than water sinks slowly, in the Dead Sea (1240 kg/m³) it is pushed up. A spherical float submerged to depth h displaces V = π h² (3r - h)/3; equilibrium is found by bisection and does not depend on g (weight and buoyancy both scale with it), but the time to settle does.

Simplifications. Quasi-static rod (no vibration), bending in the plane of rod axis and line; massless line without its own drag; the fish is a rigid body with a fixed behaviour script, no hook-pull-out; line stretch is elastic with no hysteresis beyond a damping term; the rod is held fixed on a clamp.