The Oscillating Chain Fountain · AI Generator #GEN-CDE7D00A

Physics · mechanics

Problem

A fascinating phenomenon occurs when a long chain is pulled vertically upward from a container: the chain forms a graceful arc above the container rim, resembling a fountain. Consider a uniform chain of total length

L = 2.0\;\text{m}

and linear mass density

\lambda = 0.05\;\text{kg/m}

initially coiled in a deep container. The chain is pulled vertically upward at constant speed

v_0 = 1.5\;\text{m/s}

by a motor at height

h = 0.8\;\text{m}

above the container rim. Due to the chain's flexibility and the upward acceleration of each link as it leaves the container, the emerging chain forms a curved "fountain" profile. At any instant, let

y

be the height above the container rim, where the chain has horizontal velocity component

v_x(y) = \sqrt{2gy}

due to the complex dynamics of the lifting process. The motor has been pulling for time

t = 0.6\;\text{s}

when it suddenly stops (but remains at the same height). At this moment, there are

x = 0.9\;\text{m}

of chain still in the container, and the chain fountain extends from the rim to the motor position. (a) Find the total kinetic energy of the chain at the instant the motor stops. (b) Find the period of small oscillations of the motor end of the chain after it stops, treating the chain as a pendulum with the rim as the pivot point. Take

g = 10\;\text{m/s}^2

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