Complete Question
The diagram for this question is shown on the first uploaded image
Answer:
The angular acceleration for first drum

The angular acceleration for the second drum is

Step-by-step explanation:
From the question we are told that
Their radius of the drum is
each
The weight is

The mass is

Their radius of gyration is

The free body diagram of a drum and its hub and 30lb and in the case the weight is connect to the hub separately is shown on the second uploaded image
The T in the diagram is the tension of the string
Now taking moment about the center of the the drum P we have

=>

Where r is the radius ,z is the radius of gyration about the center O , M is the mass of the drum including the hub, and
is the angular acceleration
Inputting

=>

Considering the force equilibrium in the vertical direction (Looking at the second free body diagram now )
The first on is

=>

Where m is the mass of the hanging block which has a value of

a is the acceleration of the hanging block
inputting values we have




So the angular acceleration for first drum

The free body diagram of a drum and its hub when the only on the string is 30lb is shown on the third uploaded image
So here we would take the moment about O

So

and

Therefore we will have

inputting values



So the angular acceleration for the second drum is
