The correct answer is A) centrifugal force. The over speed tripping device installed on an auxiliary turbine is automatically actuated by centrifugal force. As the turbine's rotational speed increases, the centrifugal force acting on the device also increases. When the rotational speed exceeds a pre-determined safe limit, the centrifugal force triggers the tripping mechanism, which automatically shuts down the turbine to prevent damage. The other options are incorrect because high back pressure (B), hydraulic pressure (C), and pneumatic force (D) are not the primary actuation mechanisms for the over speed tripping device on an auxiliary turbine. These factors may play a role in the overall turbine control system, but they do not directly trigger the over speed tripping device.