Classical Mechanics: Lagrangian of a falling yo-yo

[NahsiN]

Homework Statement

A uniform circular cylinder (a yo-yo) of radius a and mass M has a string wrapped around
it that can unwind without slipping. The yo-yo moves in a vertical straight line and the
straight part of the string is vertical as well. The other end of the string is fastened to a
support that has upward displacement h(t) at time t. Here h(t) is a prescribed function, not
a degree of freedom.
a. Take the rotation angle φ of the yo-yo as a generalized coordinate and find Lagrange’s equation.
b. Find the acceleration of the yo-yo. What must the support’s acceleration h(t) be so
that the centre of the yo-yo can remain at rest?
c. Suppose the system starts from rest. Find an expression for the total energy E = T+V at time t, in terms of h and h.

Homework Equations

L=T-V
T=$$\frac{1}{2}$$M$$\dot{y}$$$$^{2}$$+$$\frac{1}{2}$$I$$\dot{\varphi}$$$$^{2}$$
Defining downwards as my positive direction: V=-Mgy
Rolling without slipping ---> a$$\dot{\varphi}$$=$$\dot{y}$$
Since the support is moving upwards given by (h(t)) ---> y=$$\widetilde{y}$$-h(t) where $$\widetilde{y}$$= the distance between the yo-yo and the support at time t. So, $$\dot{y}$$=$$\dot{\widetilde{y}}$$-$$\dot{h(t)}$$

The Attempt at a Solution

With these equations and treating $$\varphi$$ as my generalized coordinate, it's easy to obtain the Lagrange's equation of motion for this system. I get $$\dot{}\dot{\varphi}$$=2/3*g/a

Now for part b) of the question, I have $$\dot{}\dot{y}$$=$$\dot{}\dot{h}$$. If this is correct, I can figure this out using the without slipping equation above.
From here on before starting part c), I am wondering why the Lagrange's equation I got is independent of h(t) (i.e its the same as if the support was fixed). And if its wrong, it has to do something with the relation between y and h(t). Any help is apprecaited

 

[Steve4Physics]

See the discussion of the same problem here: https://www.physicsforums.com/threads/lagrange-yo-yo-with-moving-support.316444/