A heavy vault door with a hydraulic closer is modelled as a mass-spring-damper: mass

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations · Simple, damped and driven oscillations

Mathematical Methods for Physics, Ch Second-order linear equations: damping, driving and resonance

Problem

A heavy vault door with a hydraulic closer is modelled as a mass-spring-damper: mass m=25 kgm = 25\,\mathrm{kg}, spring stiffness k=100 N m−1k = 100\,\mathrm{N\,m^{-1}}, damping coefficient b=110 kg s−1b = 110\,\mathrm{kg\,s^{-1}}, obeying mx¨+bx˙+kx=0m\ddot x + b\dot x + kx = 0 for the door's displacement xx from closed. The door is released from x(0)=0.15 mx(0) = 0.15\,\mathrm{m} open while already swinging shut at x˙(0)=−0.20 m s−1\dot x(0) = -0.20\,\mathrm{m\,s^{-1}}. ((a)) Find the two roots of the auxiliary equation and confirm the motion is overdamped. ((b)) Fit the two constants of the general solution to the stated initial conditions. ((c)) Find xx at t=0.50 st = 0.50\,\mathrm{s}, and state whether the door overshoots the closed position (x=0x=0) before then. ((d)) Keeping x(0)=0.15 mx(0) = 0.15\,\mathrm{m}, find the smallest inward starting speed for which the door would pass the closed position, and say why the actual push does not.

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