A bead is moving without friction on a bent wire (see figure above) under the constraint z = 2r², expressed in cylindrical coordinates. The wire rotates with constant angular velocity ω about the z-axis. Gravity is acting in the negative z-direction.

a) Find the constraint for the azimuthal angle ϕ.  

b) Classify the constraints in terms of holonomic/non-holonomic and rheonomic/scleronomic.

 c) Write down D’Alembert’s equation for the bead in Cartesian coordinates. 

d) Express the cartesian coordinates as well as the virtual displacements δx, δy, and δz of the bead in cylindrical coordinates. What is the number of degrees of freedom of the bead?