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In lecture 27, particles angular impulse and momentum, in example 2 we include the bullet in the angular momentum equation, even though at first it is moving with a linear velocity only. In tut 12, question 3, we don't include the carriage in the angular momentum equation, even though it is a part of the system. Is this difference because with the bullet, after impact it will also have an angular velocity so we include it but in tut 12 Q3 the carriage never has an angular velocity so even though it is a part of the system we don't include it in the angular momentum eqn?

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Nope, the positive direction does not matter. As long as these are different rigid bodies and we are using the force or momentum equation on them, you can choose any direction you like.

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Why are we allowed to make the positive directions different for A and B? Doesn't that affect the final answer

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In the memo we take the sum of the moments about the centre of gravity of the rod, G, but why would it be wrong to take the sum of the moments about O, the center of the loop, because surely the rod and loop will both have the same alpha value and that way we can avoid even calculating N?

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The problem with taking the moments about O, the centre of the loop (or hoop), is that you need to then use the form of the equation sum Mp = Ia + magd. So it cancels out N but introduces another term which is possible to work out since you need the vertical component of aG - which is aG/O - and then d = r/2. You can do it this way.

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The alpha that was worked out is the alpha of the links. The platform stays horizontal so Ia is zero

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why is the Ia term not included in taking the sum of the moments about A? We just worked out alpha to be non-zero and now are trying to find the force at B at this non-zero value of alpha.

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Since r = 1m, leaving it out of that exact equation will give you the correct answer!

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what value of r was used in this solution to find alpha? I keep getting the wrong answer no matter what value of r I sub into this exact equation.

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FBD nearly correct but use Table D/4 to locate G (semicylindrical shell).
You do need theta.
APOLOGIES, theta was not initially included in the tutorial question but theta = 45 deg has now been added.

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Is this FBD correct ? If so is the angle theta not needed?