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Whenever an object is moving downward we take the weight of the object to be greater than the tension in the string, and when the weight is moving upward the tension is greater than the weight we use this concept while writing the equations.
Yes, T is equal in both cases for both objects are attached to the same string and the string used is usually a light inextensible string which exerts same forces on both masses attached.
Yes, T is equal in both cases for both objects are attached to the same string and the string used is usually a light inextensible string which exerts same forces on both masses attached.