Final answer:
To calculate the moment of inertia (I) using the given variables, use the equation I = (mg r) / α, which is derived from the relationship T = Iα and connects torque, moment of inertia, and angular acceleration.
Step-by-step explanation:
To calculate the moment of inertia (I) in terms of angular acceleration (α), the mass attached to the string (m), the radius of the middle pulley groove (r), and the acceleration due to gravity (g), we need to start with the relationship that connects torque (T), moment of inertia (I), and angular acceleration (α). This relationship is given by the equation T = Iα. Knowing that torque can also be expressed as T = mg r, where m is the mass being accelerated by gravity (g) at a distance r from the axis of rotation, we can set the equations equal to each other to solve for I:
mg r = Iα
Rearranging this to solve for I gives us:
I = (mg r) / α
It should be noted that the radius (r) used is that of the pulley around which the string is wound because it determines the lever arm of the torque. Also, this calculation assumes that the object's entire mass is concentrated at the radius r, which is a simplification for more complex bodies where the mass is distributed at varying distances from the rotational axis.