Answer to Question #236182 in Classical Mechanics for alwande hadebe

Question

A block of mass m is attached to a spring, with spring constant k, and a dashpot. The dashpot provides a retarding force with damping rate b and proportional to the velocity ˙x. When the block is a distance d0 from the left wall the spring is relaxed. When extended beyond d0 and released the block exhibits damped oscillatory motion. You may assume that gravitational effects are negligible. (a) Show that the equation of motion of this sytem has the form x¨ + 2γx˙ + ω 2 0x = 0 and identify the variables, γ and ω0. (b) Consider the case when γ < ω (under-damping). The block is now pulled to the right a distance x = a, from its equilibrium position, and at time t = 0 it is released from rest. Show that the subsequent motion is given by, x(t) = ae−γt  cos ω1t + γ ω1 sin ω1t  where ω1 = p ω 2 0 − γ 2. (c) At what time does the block first change its direction of motion after being released?

Answer 1

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