Introductory application of the Newton Euler equations to a composite body

[Andrew1234]

Homework Statement

The thin uniform bar 𝐴𝐵 of mass 𝑚 and length 𝐿, with an attached uniform
disk of mass 2𝑚 and radius L/4 hangs from a wheel at 𝐴, which rolls
freely on the horizontal bar 𝐷𝐸.

Relevant Equations

Newton Euler equations

α is the second derivative of angle and w is the first derivative

In the free body diagrams the only force on A is the normal force since it is only constrained not to move vertically.
Have I drawn the free body diagram and kinetic diagram correctly?

By relating the accelerations of the centers of mass of the bar and disk to that of the wheel, which can only move horizontally, I got two different expressions for the horizontal acceleration of A. The bar and disk have no horizontal acceleration as there is no horizontal force. Can anyone explain why the two expressions for horizontal acceleration are inconsistent?

 

[BvU]

HI,

It says $A$ rolls on $DE$ so there are forces of constraint acting

 

[wrobel]

"rolls freely" means without friction thus the reaction force has vertical component only.
In this system horizontal component of the momentum is conserved and the total energy is conserved. That is enough to obtain system of equations of motion

 

[BvU]

What makes it roll instead of slide ?

 

[wrobel]

to roll by massless wheel is the same as to slide