{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"frontierscience","formal_name":"FrontierScience","introduction":"専門的な科学課題を解く能力を評価するベンチマークです。公開データはolympiadとresearchに分かれ、競技問題と研究課題を区別して扱います。\n\nFrontierScience evaluates the ability to solve expert-level scientific tasks. Its public data separates olympiad and research problems so that competition and research tasks can be examined independently.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/openai/frontierscience","indexing_mode":"noindex"},"task_id":"8514eb09-d1c3-5811-9533-09ab0426676c","task_key":"olympiad--test--84bba88d~2d2aee~2d4fcb~2dbaab~2d4c6eea17b0d8","task_revision_id":"1","upstream_id":"84bba88d-2aee-4fcb-baab-4c6eea17b0d8","short_description":"Consider a ratchet, a device shaped like a disk, but with notches around the…","config":"olympiad","split":"test","body":"{\"problem\":\"Consider a ratchet, a device shaped like a disk, but with notches around the edge. The ratchet is prevented from rotating backward by a pawl resting against one of the notches. The pawl is held against the notch by a spring. The ratchet can rotate forward. As it does so, it gradually lifts the pawl and compresses the spring. The pawl continues rising and the spring continues compressing until the ratchet has advanced one notch forward, at which point the pawl falls off the edge of the notch, releasing the elastic energy in the spring. The elastic energy in the spring just before the pawl falls of the edge of a notch is \\\\\\\\(\\\\\\\\epsilon\\\\\\\\), and the elastic energy in the spring just after the pawl falls off the edge of a notch is zero.\\n\\nThe pawl passes to a new notch every time the ratchet rotates forward by some angle \\\\\\\\(\\\\\\\\theta\\\\\\\\). If some influence causes the pawl to rise until the potential energy in the spring is greater than \\\\\\\\(\\\\\\\\epsilon\\\\\\\\), the pawl can rotate backwards.\\\\\\n\\\\\\nThe ratchet is connected to vanes via an axle. The ratchet, vanes, and axle rotate rigidly. Assume the device is small enough that when the vanes are surrounded by a gas, the impulse generated by the collision of a single gas molecule with the vanes can turn the vanes.\\n\\nAlthough the collisions of gas molecules tend to push on the vanes in either direction with equal probability, as long as the spring is not compressed with elastic energy greater than or equal to \\\\\\\\(\\\\\\\\epsilon\\\\\\\\), the pawl only allows the ratchet, and therefore the vanes, to rotate forward.\\n\\nNext, let's assume the spring of the ratchet is in thermal reservoir 2 at temperature `\\\\( T_2 \\\\)` , while the vanes are in thermal reservoir 1 at temperature `\\\\( T_1 \\\\)` , and the two reservoirs are not in thermal contact, aside from any energy that may flow through the vanes, ratchet, and pawl. Furthermore, a weight is tied to a rope hanging from the middle of the axle, so that as the ratchet turns forward, the weight is lifted against gravity. The weight causes a torque of magnitude `\\\\( \\\\tau\\\\)` to be exerted on the axle, measured relative to the center of the axle, which is also the axis about which the ratchet, axle, and vanes can rotate. The torque is in the direction that rotates the axle, ratchet, and vanes backwards.\\\\\\n\\\\\\nSuppose that \\\\\\\\(T_1 > T_2\\\\\\\\) and \\\\\\\\(\\\\\\\\tau = 0\\\\\\\\). Assume the heat capacities of the two reservoirs are \\\\\\\\(c_1\\\\\\\\) and \\\\\\\\(c_2\\\\\\\\) for reservoirs 1 and 2 respectively. Find an equation for \\\\\\\\(\\\\\\\\Delta S\\\\\\\\), the increase in entropy of the system as it goes from this initial state to its final equilibrium state. Give your answer in terms of \\\\\\\\(c_1, c_2, T_1,\\\\\\\\) and \\\\\\\\(T_2\\\\\\\\). Do not include any other variables.\\n\\nThink step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.\",\"subject\":\"physics\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/openai/frontierscience","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}