a particle of mass rest on a smooth plane. the plane is raised to an inclination

Question

a particle of mass rest on a smooth plane. the plane is raised to an inclination angle at a constant rate (=0 at t=0), causing the particle to move down the plane. Determine the motion of the particle by choosing r and as generalized coordinates. You may use any classical mechanics text for studying the motion of driving oscillator. techniques commonly used in solving inhomogeneous second order differential equation encountered in a driven oscillator problem will be useful here. This system is not oscillatory but mathematically similar to driven oscillator problem





thanks

Explanation / Answer

acceleration of the particle. a=-g*sin0. where 0 be theta. we also have that a=r''. so that r''=-g*sin0. where 0=w*t. where w be the angular velocity of the plane. so that r''=-g*sin(wt). so we have that dv/dt=-g*sin(wt). so dv=-g*sin(wt)*dt at t=0. v=0. so v=g/w*(cos(wt)-1). so that dr/dt=(g/w)*(cos(wt)-1). so that r=(g/w)*(-sin(wt)/w-t)+r0 r=-(g/w^2)*sin(wt)-gt/w+r0

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