angular speed, torque, angular speed and I will add angular inertia (moment of inertia) change with radius.

You are told that the radius changes with time.

Clearly the radius is the common link between changes to thes evariables and changes with time (the question)

So what do you know (mathematically) about the link between radius and these quantities?

Is there any one can help me to make a relationship between the Torque and time ? i will be thankful

Best regards

Thanks for replay , i have tried to relate these quantities but i failed , there are too many variables changing at the same time , angular velocity , radius of the reel , linear velocity of the reel ,and also we have two reels in the tape , so i think we need to model this problem as a differential equation and then solve it , but i am still missing something , any idea for help ?

Best Regards

I think you are in danger of overthinking.

All that is needed is the relationship between angular velocity and radius,

and the relationship between torque and radius.

the tape speed is

The second reel, for example, is irrelevant (to the question being answered)

This is essentially the same as a gear wheel problem.

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