Why is the tension in the string different in the diagram below?
Why can't we use the same tension for both the strings as we usually do in Atwood system?
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This is a system in which the various components are acceleration which includes the angular acceleration of the pulley.
Taking the pulley as the system then in order that it undergoes an angular acceleration about its centre of mass there must be an external torque acting on it.
That torque can only be be provided by the string having a different tension on either side of the pulley.