All Problems Mechanical Oscillations
Problem 4.84
A particle of mass \(m\) can perform undamped harmonic oscillations due to an electric force with coefficient \(k\). When the particle was in equilibrium, a permanent force \(F\) was applied to it for \(\tau\) seconds. Find the oscillation amplitude that the particle acquired after the action of the force ceased. Draw the approximate plot \(x(t)\) of oscillations. Investigate possible cases.
4.84. The motion equations and their solutions: \[ \begin{array}{ll} t \leqslant \tau, \ddot{x}+\omega_{0}^{2} x=F / m, & x=\left(1-\cos \omega_{0} t\right) F / k, \\ t \geqslant \tau, \ddot{x}+\omega_{0}^{2} x=0, & x=a \cos \left[\omega_{0}(t-\tau)+\alpha\right], \end{array} \] where \(\omega_{0}^{2}=k / m, a\) and \(\alpha\) are arbitrary constants. From the continuity of \(x\) and \(x\) at the moment \(t=\tau\) we find the sought amplitude: \[ a=(2 F / k)\left|\sin \left(\omega_{0} t / 2\right)\right| \]