{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"scicode","formal_name":"SciCode","introduction":"SciCode evaluates the ability to solve scientific research problems through code. Problems are decomposed into subproblems; this dev import preserves the relationships between 15 parent problems and 50 subproblems.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/SciCode1/SciCode","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"fbb65551-2f68-5fcd-af48-e106fe7a12f3","task_key":"dev--831ff049-2748-5e6d-ba9b-f8252555cd23--78~2e1","task_revision_id":"3","upstream_id":"78.1","short_description":"The motion of a forced, damped single pendulum can be described by the…","config":"","split":"dev","body":"{\"step_background\":\"Background\\n\\nA single pendulum consists of:\\n- A mass $m$ attached to the end of a string or rod of length $L$.\\n- The pivot point is frictionless.\\n- There is a damping force proportional to the angular velocity with a damping coefficient $\\\\beta$.\\n- An external driving force $A \\\\cos(\\\\alpha t)$ oscillates with time.\\n\\n(a) Identify the Forces and Torques\\n\\nThe forces acting on the pendulum bob are:\\n- Gravitational force $mg$ acting downward.\\n- Tension in the string (which does not do work as it acts along the string).\\n- Damping force proportional to the angular velocity $-\\\\beta \\\\frac{d\\\\theta}{dt}$.\\n- External driving force $A \\\\cos(\\\\alpha t)$.\\n\\nThe torque $\\\\tau$ around the pivot due to the gravitational force, damping force, and driving force is given by:\\n\\n$$\\n\\\\tau = -mgL \\\\sin(\\\\theta) - \\\\beta L \\\\frac{d\\\\theta}{dt} + A \\\\cos(\\\\alpha t)\\n$$\\n\\nThe moment of inertia $I$ for a point mass $m$ at a distance $L$ from the pivot is:\\n\\n$$\\nI = mL^2\\n$$\\n\\n(b) Apply Newton's Second Law for Rotation\\n\\nUsing Newton's second law for rotation $\\\\tau = I \\\\alpha$:\\n\\n$$\\n-mgL \\\\sin(\\\\theta) - \\\\beta L \\\\frac{d\\\\theta}{dt} + A \\\\cos(\\\\alpha t) = mL^2 \\\\frac{d^2\\\\theta}{dt^2}\\n$$\\n\\n(c) Simplify the Equation\\n\\nDividing both sides by $mL^2$:\\n\\n$$\\n-\\\\frac{g}{L} \\\\sin(\\\\theta) - \\\\frac{\\\\beta}{mL} \\\\frac{d\\\\theta}{dt} + \\\\frac{A}{mL^2} \\\\cos(\\\\alpha t) = \\\\frac{d^2\\\\theta}{dt^2}\\n$$\\n\\nRearranging the terms:\\n\\n$$\\n\\\\frac{d^2\\\\theta}{dt^2} + \\\\frac{\\\\beta}{mL} \\\\frac{d\\\\theta}{dt} + \\\\frac{g}{L} \\\\sin(\\\\theta) = \\\\frac{A}{mL^2} \\\\cos(\\\\alpha t)\\n$$\\n\\n(d) Converting to First-Order ODEs\\n\\nTo use numerical methods like RK4, we need to convert this second-order ODE into a system of first-order ODEs by defining:\\n\\n$$\\n\\\\frac{d\\\\theta}{dt}= \\\\omega\\n$$\\n\\n\\nThen, the first-order ODEs for this damped, forced single pendulum system becomes:\\n\\n$$\\n\\\\frac{d\\\\omega}{dt} = -\\\\frac{g}{L} \\\\sin(\\\\theta) - \\\\frac{\\\\beta}{mL} \\\\omega+ \\\\frac{A}{mL^2} \\\\cos(\\\\alpha t)\\n$$\\n\\n(e) Define The State Vector\\n\\nThe state vector of this forced, damped pendulum is:\\n\\n$$\\n\\\\mathbf{y} = \\\\begin{pmatrix} \\\\theta \\\\\\\\ \\\\omega \\\\end{pmatrix}\\n$$\",\"step_description_prompt\":\"The motion of a forced, damped single pendulum can be described by the second-order differential equation. Assuming the damping force is proportional to the angular velocity with coefficient $\\\\beta$ and the external driving force *$A \\\\cos(\\\\alpha t)$* oscillates with time ($A$: drive amplitude, $\\\\alpha$:drive frequency). Define the differential equations in python for this forced, damped Pendulum. The solution of this differential equation, $ y=[\\\\theta, \\\\omega]^T$, represents the state vector of the system, which includes the angle *$\\\\theta$* and the angular velocity *$\\\\omega$*. $\\\\theta$ is measured in radians and $\\\\omega$ in radians per second.\"}","display_format":"scicode-step","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/SciCode1/SciCode","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}