An Atwood machine consists of two blocks of masses \(m_1\) and \(m_2\) (where \(m_2 > m_1\)) connected by a light string that passes over a uniform solid cylindrical disk pulley of mass \(M\) and radius \(R\). The pulley is mounted on a frictionless horizontal axle, and the string does not slip on the pulley as the system accelerates from rest. What is the difference between the tensions in the two vertical segments of the string, \(T_2 - T_1\), in terms of the given quantities and fundamental constants?