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The working principle of Borden tube based Pressure gauges is on Hooks Law Principle.
As we know that when the pressure applies on the surface of the body it becomes the source of stress on the body, their are two sides of the Borden tube inner & the outer. The outer side of the Borden tube has area greater then inner side, that's why the pressure of the outer side less then the inner side so the Borden tube moves toward the outer side.