All Problems Mechanical Oscillations
Problem 4.42
A particle of mass \(m\) moves in the plane \(x y\) due to the force varying with velocity as \(\mathbf{F}=a(\dot{y} \mathbf{i}-\dot{x} \mathbf{j}),\) where \(a\) is a positive constant, i and j are the unit vectors of the \(x\) and \(y\) axes. At the initial moment \(t=0\) the particle was located at the point \(x=y=0\) and possessed a velocity \(\mathbf{v}_{0}\) directed along the unit vector \(\mathbf{j} .\) Find the law of motion \(x(t), y(t)\) of the particle, and also the equation of its trajectory.
4.42. Let us write the motion equation in projections on the \(x\) and \(y\) axes: \(\ddot{x}=\omega \dot{y}, \ddot{y}=-\omega x,\) where \(\omega=a / m\) Integrating these equations, with the initial conditions taken into account, we get \(x=\left(v_{0} / \omega\right)(1-\cos \omega t), y=\left(v_{0} / \omega\right) \sin \omega t .\) Hence \(\left(x-v_{0} / \omega\right)^{2}+y^{2}=\left(v_{0} / \omega\right)^{2}\). This is the equation of a circle of radius \(v_{0} / \omega\) with the centre at the point \(x_{0}=v_{0} / \omega, y_{0}=0\)