The Wobbling Coin Tower · AI Generator #GEN-479A1A21

Physics · mechanics

Problem

A street performer builds a tower by stacking identical coins on a slowly rotating turntable. Each coin has mass m=5.0  gm = 5.0\;\text{g}, radius R=12  mmR = 12\;\text{mm}, and thickness t=2.0  mmt = 2.0\;\text{mm}. The turntable rotates at a constant angular velocity ω=0.50  rad/s\omega = 0.50\;\text{rad/s}. The performer discovers that when the tower reaches a certain height HH, the entire tower begins to wobble and tip over due to the flexibility of the coin stack. The critical condition occurs when the restoring torque from gravity equals the destabilizing torque from the centrifugal effect. Treating the tower as a uniform rod of length HH pivoting at its base, find the maximum stable height HmaxH_{\text{max}} of the coin tower. Assume the center of mass of the tower is displaced horizontally by a small distance δ=1.0  mm\delta = 1.0\;\text{mm} from the rotation axis due to imperfect stacking. Take g=9.8  m/s2g = 9.8\;\text{m/s}^2.

Browse the problem library