The Spinning Coin's Death Spiral · AI Generator #GEN-ECB772B1

Physics · mechanics

Problem

A coin of radius R=12.5  mmR = 12.5\;\text{mm}, mass m=5.7  gm = 5.7\;\text{g}, and moment of inertia I=12mR2I = \frac{1}{2}mR^2 is spun on a horizontal table with initial angular velocity ω0=180  rad/s\omega_0 = 180\;\text{rad/s} about a vertical axis through its rim (not its center). Due to air resistance, the coin experiences a torque τ=−kω2\tau = -k\omega^2 about the contact point, where k=3.2×10−9  N⋅m⋅s2k = 3.2 \times 10^{-9}\;\text{N}\cdot\text{m}\cdot\text{s}^2. As the coin slows down, it gradually moves inward in a spiral pattern toward the center of the initial circular motion. Find the total distance the coin travels along its spiral path before coming to rest. Assume the coin maintains contact with the table throughout and that rolling friction is negligible compared to air resistance.

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