Coin Pusher Lab

Tokens on a shelf, a moving pusher plate and side gutters: a 2-D granular model of friction, jamming and what a coin is really worth.

Presets

Feeding the machine

Pusher

Friction (Coulomb mu)

Machine and world

Display

Coins in and out (cumulative)

droppedpaid outlost to gutters

Force on the pusher face (N)

mean over 40 msslow average

Pile packing along the shelf (area fraction)

packing fractionpusher face

Ledger

1 - Is the pusher force really mu m g per coin?

A block of N coins is pushed at 5 cm/s along the shelf with a plain face (no plate). In steady sliding every coin feels a friction force μ m g, so the face must supply F = μ N m g (μ = 0.35 here, gravity from Setup). Rows show the measured mean force, the prediction and the ratio; the last row adds friction on the side walls.

2 - Expected payout with a confidence interval

Five independent runs of the current Setup (different random jitter and drop positions): 40 coins, one every 1.5 s, starting from the Setup's pile. Coins are conserved - the table checks it - so the interesting numbers are how many come back, how many vanish into the gutters, and how the net result evolves; the band is the 95 % t-interval across runs.

3 - Does where you drop the coin matter?

Centre, random and hugging the left wall: four runs each of 30 coins (one per 1.5 s). Compared by the share of exiting coins that fall into the gutters; with a uniform flow the gutter share should equal 2 g_w / W.

The science

Model. A top view of a single layer of discs (diameter 34 mm, 7.35 g by default, aluminium-like). Each token has position, velocity, angle and spin. Equations of motion are integrated with semi-implicit Euler at dt = 1 ms with a cell grid for neighbours. Contact normal force F_n = k δ - c v_n (linear spring-dashpot, k = 1000 N/m); contact friction F_t = -μ F_n v_t/√(v_t²+v_e²) with the contact slip speed including spin (a regularised Coulomb law, v_e = 5 mm/s).

Floor friction. A token sliding on the shelf feels -μ_s m g v_rel/|v_rel|; on the moving plate the relative velocity is taken against the plate. Spin is damped by the torque 2/3 μ m g r of a disc contact patch.

Stick or slip on the plate. The plate has sinusoidal motion y = (S/2)(1 - cos 2πt/T): peak acceleration a_max = (S/2)(2π/T)². A token rides with the plate while a_max < μ_p g and slips otherwise; the Setup tab shows which case applies (on the Moon the grip is six times weaker).

The ratchet. When the plate retracts, tokens near the back wall are held by the jammed pile while the plate slides underneath; tokens that end up in front of the retreating face are then on the fixed shelf, and the next forward stroke pushes them on. Only a dense pile jams, so payouts need a loaded machine - an empty one pays nothing.

Pusher force. Sliding a block of N tokens needs F = μ_s N m g; wall friction adds a Janssen-type term because pushing the pile presses it sideways against the walls. Experiment 1 checks the first claim numerically.

Expected payout. Tokens are conserved: dropped + start pile = paid + gutters + still on the shelf. Over a long time the payback per coin is therefore 1 minus the gutter share, and for a uniform stream of exits the gutter share is 2 g_w/W. A machine pays back the bank in bursts, which is why individual runs scatter and the experiment reports a confidence interval over replicates (t-distribution, n - 1 degrees of freedom).

Simplifications. One layer only - real machines pile coins two to four deep, which adds weight on the plate and avalanches; coins that would land on a full shelf wait in a queue on top of the pile and slide in when space appears; no prizes; walls and plate are perfectly rigid.