A camera's autofocus mechanism is designed to be critically damped

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 camera's autofocus mechanism is designed to be critically damped, with ω0=50 rad s−1\omega_0 = 50\,\mathrm{rad\,s^{-1}}, so that its lens displacement xx from the focus position obeys x¨+2ω0x˙+ω02x=0\ddot x + 2\omega_0\dot x+\omega_0^2 x=0. After a fast correction the lens is at x(0)=4.0 mmx(0) = 4.0\,\mathrm{mm} and moving at x˙(0)=−0.30 m s−1\dot x(0) = -0.30\,\mathrm{m\,s^{-1}}. ((a)) Write the general critically damped solution and fit its two constants to the stated initial conditions. ((b)) Find the time at which the lens passes through the focus position (x=0x=0). ((c)) Find the lens's velocity at that instant.

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