The Shop Sign on Two Chains
A shop sign hangs from two chains at different angles. Before touching anything, it is worth predicting which chain is working harder — the steep one or the one closer to the horizontal. Set the two angles, and the sign settles at the point where the chains meet, sliding towards the steeper one, which is also the one under the greater tension.
Como usar esta simulação
- Set the left chain's angle to the horizontal, anywhere from 15° to 70°
- Set the right chain's angle independently, and watch the sign slide towards the steeper one
- Change the mass of the sign from 2 kg to 20 kg
- Use the camera bar for a street view or a close look at the ring
O que observar
- The steeper chain always carries the greater tension, whatever the mass
- Equal angles give equal tensions, and the sign hangs squarely between the hooks
- Six kilograms on two cables at 30° gives 58.8 N in each — the lamp's whole weight
- An 8 kg sign at 30° and 55° gives 45 N and 68 N, and the 55° chain is the loaded one
- The two horizontal components stay equal and opposite at every setting
- Flattening both chains towards 15° pushes the tension past twice the weight
A física por trás
Three forces act at the ring the two chains meet on: the two tensions and the weight of the sign hanging below. Resolving along two perpendicular axes gives one equation each. Across: T_L cos θ_L = T_R cos θ_R, since the horizontal components must cancel. Upward: T_L sin θ_L + T_R sin θ_R = W, since the vertical components together carry the weight. Solving the pair gives T_L = W cos θ_R / sin(θ_L + θ_R) and T_R = W cos θ_L / sin(θ_L + θ_R). The steeper chain has the larger sine, so it lifts a bigger share and carries the larger tension. With both chains at the same angle the problem collapses to 2T sin θ = W, and as θ falls towards the horizontal sin θ collapses with it and the tension runs away.