Chain Sliding Off Table
A uniform chain with linear density λ = M/L lies on a horizontal table with a smooth vertical edge. The hanging segment of length s pulls with weight λgs; the segment on the table (length L−s) contributes normal force and friction μλg(L−s). Treating the whole chain as one system, the net horizontal acceleration along the slide is a = (g/L)(s − μ(L−s)) once static friction is overcome (s > μ_s(L−s)).
Who it's for: Intro mechanics; capstan/chain puzzles and friction demos.
Key terms
- static friction
- kinetic friction
- linear mass density
- critical overhang
- sliding
How it works
A uniform chain lies on a horizontal table with a smooth edge. The hanging part pulls with weight proportional to its length; the part on the table presses down and can exert friction. The model treats the whole chain as one mass: net force is λg(s − μ(L−s)) along the slide, giving a = (g/L)(s − μ(L−s)) while moving. Static equilibrium holds until s exceeds μ_s times the length still on the table.
Key equations
Slip when: s > μ_s (L − s) ⇒ s_crit = μ_s L / (1 + μ_s)
a = (g/L)(s − μ_k(L − s)), ṡ = v, s̈ = a
Frequently asked questions
- Why does acceleration increase as more chain hangs?
- The driving term grows with s while friction on the table shrinks as (L−s). Both effects make the net force larger until the last chain leaves the table.
- Is this model exact for a real chain?
- The edge is assumed smooth; the chain bends as a rigid L-shape. Real chains have edge curvature and internal friction, but the 1D balance captures the main competition between weight and table friction.
More from Classical Mechanics
Other simulators in this category — or see all 71.
Block Stack & Friction
Pull the bottom block in a vertical stack: spring-like links with friction caps show who slips — compare with rigid-stack estimates.
Bicycle Stability (2D)
Side view: roll dynamics with fork trail and gyroscopic wheel torques — see speed, trail, and ω = v/R vs lean.
Spring Pendulum
2D elastic pendulum: swing and spring stretch together — energy swaps between modes; try chaotic-looking presets.
Bridge Resonance (1-D mode)
Damped modal oscillator with harmonic drive: sweep ω near √(k/m) — presets for cadence-like and low-ζ peaks.
Rising Bubble (Archimedes)
Ideal-gas bubble expands as it rises (p = p_atm + ρgy); buoyancy vs weight + Stokes drag — depth and velocity graphs.
Belt Drive & Slip
Two pulleys: tension difference limited by μ and wrap θ (e^{μθ}); load torque vs τ_max and simple slip on ω₂.