All Problems Mechanical Oscillations
Problem 4.59
Two balls with masses \(m_{1}=1.0 \mathrm{~kg}\) and \(m_{2}=2.0 \mathrm{~kg}\) are slipped on a thin smooth horizontal rod (Fig. 4.19). The balls are interconnected by a light spring of stiffness \(x=24 \mathrm{~N} / \mathrm{m} .\) The lefthand ball is imparted the initial velocity \(v_{1}=12 \mathrm{~cm} / \mathrm{s} .\) Find: (a) the oscillation frequency of the system in the process of motion; (b) the energy and the amplitude of oscillations.
4.59. (a) \(\omega=\sqrt{x / \mu}=6 \mathrm{~s}^{-1}\) (b) \(E=1 / 2 \mu v_{1}^{2}=5 \mathrm{ml} J, a=v_{1} / \omega=\) \(=2 \mathrm{~cm} .\) Here \(\mu=m_{1} m_{2} /\left(m_{1}+m_{2}\right)\)