Dynamics - Pulley System with Rotating Rod

[bonannic]

Homework Statement

At the instant shown, the rod R is rotating about its centre of rotation with ω=3.8rad/s.

mA=10kg;

The pulley, with m_P=8.7kg and R_P=0.2m, may be modeled as a uniform disc.

The rod, with m_R=4.1kg and L=0.8m, may be modeled as a thin beam rotating about one end.

g=9.8m/s ².

What is the magnitude of the acceleration of point B at this instant?

Homework Equations

ΣF=ma (N2) ΣM=Iα (Eulers equation)

The Attempt at a Solution

I_P=(1/2)MR²
I_{Rod at centre of rotation}=(1/3)ML²

I defined upwards and anticlockwise to be positive and thus derived the following equations:
ΣF_A=T_A-m_Ag=m_Aa_A
ΣM_{P at centre}=R_pT_A-R_pT_B=I_Pα_P
ΣM_{Rod at end}=-LT_B+(1/2)LM_Rg=I_Rα_R
where T_A= Tension force acting between A and pulley and T_B=Tension force acting between rod and pulley

I then found these constraints on a_B in terms of a_A,α_P,α_R
-a_B=a_A
5a_B=α_P
(-5/4)a_B=α_R
assuming that a^B is acting upwards

Then, by subbing a_B into the three original equations, I got the following system of equations:
T_A+m_Aa_B=m_Ag
R_pT_A-R_pT_B-5I_Pa_B=0
-LT_B+(5/4)I_Ra_B=-(1/2)LM_Rg

However, when I solve this system of linear equations I get the wrong answer. I have a feeling this is because I ignored the angular velocity of the rod but I can't see that would affect the acceleration of B.

 

[Orodruin]

If you ignored everything (gravity and tension) except for the fact that the rod is rotating. Would B have zero acceleration?

 

[bonannic]

The question does not specify. All the information in the problem statement is all the information that the question gives.

 

[Orodruin]

It is not a question about the problem, it is a question to you.

 

[bonannic]

Ah ok. So the end of the rod would be accelerating towards the pivot point as well.
I took this into consideration and got the right answer. I guess I should have studied the end of the rod more closely. Thank you so much for your help :)