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Mechanical Oscillations

Problem 4.84

A particle of mass mm can perform undamped harmonic oscillations due to an electric force with coefficient kk. When the particle was in equilibrium, a permanent force FF was applied to it for τ\tau seconds. Find the oscillation amplitude that the particle acquired after the action of the force ceased. Draw the approximate plot x(t)x(t) of oscillations. Investigate possible cases.

Reveal Answer
4.84. The motion equations and their solutions: t⩽τ,x¨+ω02x=F/m,x=(1−cos⁡ω0t)F/k,t⩾τ,x¨+ω02x=0,x=acos⁡[ω0(t−τ)+α], \begin{array}{ll} t \leqslant \tau, \ddot{x}+\omega_{0}^{2} x=F / m, & x=\left(1-\cos \omega_{0} t\right) F / k, \\ t \geqslant \tau, \ddot{x}+\omega_{0}^{2} x=0, & x=a \cos \left[\omega_{0}(t-\tau)+\alpha\right], \end{array} where ω02=k/m,a\omega_{0}^{2}=k / m, a and α\alpha are arbitrary constants. From the continuity of xx and xx at the moment t=τt=\tau we find the sought amplitude: a=(2F/k)∣sin⁡(ω0t/2)∣ a=(2 F / k)\left|\sin \left(\omega_{0} t / 2\right)\right|
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