3. Mass on Inclining Plane. [10 pts.]
A mass \( m \) rests on a horizontal, frictionless plane. The plane is inclined at a constant rate \( \alpha \), i.e., \( \dot{\theta}=\alpha \), which causes the particle to slide down the plane.
(a) Determine the Lagrangian \( \mathcal{L} \) for the mass.
(b) Write Lagrange's equations for the \( \mathcal{L} \) obtained in part (a).
(c) Solve the equation of motion to determine the position down the plane of the mass as a function of time.