(Solved): 6. A particle of mass \( m \) moves across the surface of a smooth, frictionless table, never leav ...

6. A particle of mass \( m \) moves across the surface of a smooth, frictionless table, never leaving the surface of the table. The particle is connected to a linear spring, with spring constant \( k \) and unstretched length \( \ell_{0} \), via a massless, inextensible cable that runs through a hole located in the middle of the table, which is denoted point \( O \). Let \( \mathcal{N}=\left\{O, \hat{n}_{1}, \hat{n}_{2}, \hat{n}_{3}\right\} \) be an inertial coordinate system, and let \( \mathcal{E}=\left\{O, \hat{\boldsymbol{e}}_{r}, \hat{e}_{\theta}, \hat{e}_{3}\right\} \) be a rotating coordinate system that tracks the location of the particle via \( \hat{\boldsymbol{e}}_{r} \), where \( \hat{e}_{3}=\hat{\boldsymbol{n}}_{3} \). Gravity acts in the \( -\hat{\boldsymbol{n}}_{3} \) direction, where \( g \) is the constant magnitude of the acceleration due to gravity. Express all vector results in coordinate system \( \mathcal{E} \). (a) Sketch a free body diagram of the particle. (b) Find the total force acting on the particle. (c) Find the inertial acceleration of the particle. (d) Find the equations of motion for the particle. (e) Give expressions for all contact forces. (f) Find the kinetic energy of the particle. Is the kinetic energy conserved? Explain. (g) Find the angular momentum of the particle about \( O \). Is the angular momentum conserved? Explain.