Solving for Force Given Rotational Inertia and Mass

[MathewsMD]

A 0.70-kg disk with a rotational inertia given by MR2/2 (M) is free to rotate on a fixed horizontal axis suspended from the ceiling. A string is wrapped around the disk and a 2.0-kg mass (m) hangs from the free end. If the string does not slip, then as the mass falls and the cylinder rotates, the suspension holding the cylinder pulls up on the cylinder with a force of:

A. 6.9N
B. 9.8N
C. 16N
D. 26N
E. 29N
ans: B

My solution:

ma = mg - F
Fr = Ia/r
mg - ma = Ia/r²

mg - ma = Ma/2

a = (mg)/(m + 0.5M)

ƩF_y = F_N - Mg - Ma

F_N = M (g + a) = ~9.77N

Please point out any errors since I really want to ensure I understood every process throughly and correctly. If you have anything to add, in terms of helpful steps or things to consider in general, please reply! :)

 

[haruspex]

You were doing well until here:

ƩF_y = F_N - Mg - Ma

Look at the free body diagram for the disk (cylinder?).
Where is there an Ma in there?

 

[zahbaz]

[ignore/]

 

[MathewsMD]

"ƩFy = FN - Mg - Ma"

Another way to ask what haruspex is getting at:
What does ƩFy equal? Meaning, do we have acceleration or are the sum of y forces zero?

Sorry, I meant to add ƩF_y = 0, isn't that a correct assumption?

 

[haruspex]

Sorry, I meant to add ƩF_y = 0, isn't that a correct assumption?

Yes, but you don't have the right contributors to ƩF_y.

 

[zahbaz]

Ohh... yeah, that's a totally correct assumption. I misread the scenario!

I got the same answer, 9.77... but not sure where you have the Ma term as haruspex mentioned. If you consider the mass is falling with F = ma, how much tension remains on the line? This is a downwards force on the cylinder.

ƩFy = FN - Mg - tension

 

[MathewsMD]

Okay, thank your for clearing that up. Yes, I used the wrong variable name but calculated it as ma was tension.

 

[haruspex]

Okay, thank your for clearing that up. Yes, I used the wrong variable name but calculated it as ma was tension.

OK, so you meant ƩF_y = F_N - Mg - m(g-a), right?
But in the next line of the OP you wrote M (g + a), which is not so easily explained?