im really confused about signs when im writing the net torque=moment of inertia*

Question

im really confused about signs when im writing the net torque=moment of inertia*angular acceleration for the disk and the sphere.

Three Masses tau disk, net = (T hoop - T sphere)R disk = I disk alpha disk = I disk a/R disk = 1/2m disk R2 disk a/R disk = 1/2 m disk R disk a (T hoop - T sphere) = 1/2 m disk a tau sphere, net = - f sphere R sphere = I sphere alpha sphere = I sphere a/R sphere a/ R sphere = 2/5 m sphere R sphere a f sphere = -2/5 m sphere a An green hoop with mass mh = 2.6 kg and radius Rh = 0.13 m hangs from a string that goes over a blue solid disk pulley with mass md = 1.8 kg and radius Rd = 0.1 m. The other end of the string is attached to a massless axel through the center of an orange sphere on a flat horizontal surface that rolls without slipping and has mass ms = 3.9 kg and radius Rs = 0.19 m. The system is released from rest What is magnitude of the linear acceleration of the hoop?

Explanation / Answer

Start by writing "F=ma" equations for each of the things that moves. Also, since some of the objects (the pulley and the orange sphere) rotate, you should write "? = I?" equations (net torque = moment of inertia

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