Rolling motion of Cylinders and Plank on a Slope

[Kingster426]

Homework Statement

A thin plank of mass M is placed centrally across two solid cylindrical rollers each
of mass m, and the system is allowed to move freely from rest without slip down a
slope of angle θ as shown in the figure below.
(a) Derive expressions for the initial accelerations of the plank and the roller.
Your solution should include the following in a systematic way: Marks
(i) Identification of the types of ensuing motions of the plank and
the rollers,
(ii) free-body, kinematic and kinetic diagrams with an appropriate
global axes system clearly shown,
(iii) statements of relevant kinematic and kinetic relationships with
reference to your chosen global axes system in (ii), and
(iv) clear, logical and systematic manipulations of the relationships
you have stated in (iii) above to arrive at the desired results.
(b) Using the results you have obtained in part (a)(iv) above, explain
whether the acceleration of the plank changes as the system
moves so that the plank is no longer in the central position.

Figure:

Homework Equations

F=Ma
M=I

The Attempt at a Solution

Correction to the last image (7) should equal a(G) not a(r)
Answers that I have are equations 7 and 8
So I am not sure if I have done this right so I am asking if someone could verify if correct explain the process that I should be using if I'm not. I have different answers to a friend doing the same question as well.
I am also unsure of what B is asking me to do.

Thanks in Anticipation

 

[Kingster426]

Relevant equations bugged out. M=I(Alpha)

 

[haruspex]

There should be a separate free body diagram for each rigid body. It should show only forces that apply directly to that body. E.g in the FBD for a roller there should not appear an Mg force; rather, a label for an unknown force or forces from the plank.
That approach would have avoided an error in your very first diagram.

Also, although the initial situation is the same for both rollers, it will not be true thereafter.

 

[Kingster426]

So like this?

 

[haruspex]

So like this?

That's better, but you seem to be making an unwarranted assumption about the force between the cylinder and the plank.

 

[Kingster426]

Vertically down on the cylinder at the point of contact instead of at an angle?

 

[haruspex]

Vertically down on the cylinder at the point of contact instead of at an angle?

That's still an assumption. At this stage you have no firm basis for saying which way the force goes at that contact. Leave it open, e.g. as a normal and a tangential component.

 

[Kingster426]

Okay I then have to draw one for the plank? Also, any thoughts/help on part 4 of the part A?

 

[haruspex]

Okay I then have to draw one for the plank? Also, any thoughts/help on part 4 of the part A?

At some point you need to write down equations based on the FBDs. I'm not sure whether this is supposed to be in part iii or part iv, but please post an attempt.