physics please help

A yo-yo with mass M and radius R is wound with a light string. Someone drops it from rest with the string attached to a metal pole. The yo-yo falls and unwinds (spins) without slipping. The moment of inertia for the yo-yo is ½ MR^2 (a disk).

a. In terms of g, find the downward acceleration of the center of mass of the yo-yo as it unrolls from the string.

b. What is the tension in the string as the disk unwinds and falls?

c.While descending, does the center of mass of the yo-yo move to the left, the right, or straight down? Explain your answer in complete sentences.

I think straight down, but why?

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  1. T = string force up
    m g = weight force down
    m a = mg - T

    (I guess the string is wound around the outside of this yo yo)
    Torque = I alpha
    T R = .5 m R^2 alpha
    but alpha = a/R
    so T = .5 m a

    so
    ma = T - .5 ma
    (3/2) m a = (1/2) m g
    a = g/3
    T = .5 m a = (1/2)m g/3
    T = mg/6

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  2. check my arithmetic
    yes, no side force so no side motion

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