A uniform solid sphere of mass M and radius R is free to rotate about a horizontal axis through its center. A string is wrapped around the sphere and is attached to an object of mass m. Assume that the string does not slip on the sphere. Find the acceleration of the object and the tension in the string.

The force diagram below shows the forces acting on the sphere and the hanging object. The tension in the string is responsible for the angular acceleration of the sphere and the difference between the weight of the object and the tension is the net force acting on the hanging object. Use Newton’s second law to obtain two equations in a and T that we can solve simultaneously. The moment of inertia is that found for a sphere and will not be derived here.
http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html

http://hyperphysics.phy-astr.gsu.edu/hbase/Newt.html