Rotation, Torque, and Rotational Dynamics Study Guide

{"type":"doc","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"{\"type\":\"doc\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q1. Pulley and Weight—Angular Velocity-Time and Acceleration-Time Graphs\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rotational Kinematics and Dynamics\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your understanding of how torque and angular acceleration affect the angular velocity of a rotating object, specifically a pulley with a weight attached.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Angular velocity (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\"): The rate of change of angular displacement.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Angular acceleration (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\alpha\"}},{\"type\":\"text\",\"text\":\"): The rate of change of angular velocity.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau\"}},{\"type\":\"text\",\"text\":\"): \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = I\\\\alpha\"}},{\"type\":\"text\",\"text\":\" where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I\"}},{\"type\":\"text\",\"text\":\" is the moment of inertia.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Visualize the scenario: The pulley starts with a positive angular velocity (counterclockwise), slows down due to a torque in the opposite direction, stops, and then reverses direction (clockwise).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recall that a constant torque produces a constant angular acceleration. Since the torque opposes the initial motion, \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\alpha\"}},{\"type\":\"text\",\"text\":\" is negative and constant.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Sketch the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\" vs. time graph: Start at a positive value, decrease linearly to zero (when the pulley stops), then continue into negative values as the pulley reverses direction.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Sketch the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\alpha\"}},{\"type\":\"text\",\"text\":\" vs. time graph: Since the torque is constant, "},{"type":"inlineMath","attrs":{"latex":"\\\\alpha"}},{"type":"text","text":" is a horizontal line below the time axis (negative and constant).\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q2. Angular Acceleration vs. Time Graph from Angular Velocity vs. Time Graph\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Graphical Analysis of Rotational Motion\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to interpret an angular velocity vs. time graph and sketch the corresponding angular acceleration vs. time graph.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Angular acceleration (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\alpha\"}},{\"type\":\"text\",\"text\":\") is the slope of the angular velocity (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\") vs. time graph.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Examine each segment of the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\" vs. time graph and determine if the slope is positive, zero, or negative.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each segment, assign a value to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\alpha\"}},{\"type\":\"text\",\"text\":\" (positive, zero, or negative) and sketch it as a horizontal line for that time interval.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the magnitudes of the slopes to determine the relative heights of the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\alpha\"}},{\"type\":\"text\",\"text\":\" segments.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Remember that a steeper slope means a larger magnitude of angular acceleration.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q3. Three-Dimensional Point Objects—Moment of Inertia About the X-Axis\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Moment of Inertia for Rigid Bodies\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to rank the moment of inertia for different arrangements of masses about a given axis.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Moment of inertia (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I\"}},{\"type\":\"text\",\"text\":\"): \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I = \\\\sum m_i r_i^2\"}},{\"type\":\"text\",\"text\":\" where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"m_i\"}},{\"type\":\"text\",\"text\":\" is the mass and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r_i\"}},{\"type\":\"text\",\"text\":\" is the distance from the axis.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify which spheres are located on the x-axis (distance to axis is zero, so they do not contribute to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I\"}},{\"type\":\"text\",\"text\":\").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each case, count the number and type of spheres (brass or aluminum) not on the x-axis.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recall that heavier spheres (brass) contribute more to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I\"}},{\"type\":\"text\",\"text\":\" than lighter ones (aluminum), and all are at the same distance from the axis.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the cases based on the total mass not on the x-axis, since all are at equal distance.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q4. Flat Objects—Moment of Inertia Perpendicular to Surface\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rotational Inertia of Planar Objects\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to compare the moment of inertia of a circular ring, a circular disc, and a square loop, all with the same mass and outer dimension, about an axis through their centers and perpendicular to their surfaces.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Moment of inertia depends on both mass and how far that mass is from the axis of rotation.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For a ring: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I_{\\\\text{ring}} = MR^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For a disc: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I_{\\\\text{disc}} = \\\\frac{1}{2}MR^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For a square loop: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I_{\\\\text{square}}\"}},{\"type\":\"text\",\"text\":\" (all mass at corners, farther than \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"R\"}},{\"type\":\"text\",\"text\":\")\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Consider where the mass is located for each object relative to the axis.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recall that the farther the mass is from the axis, the greater the moment of inertia.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the effective distances for each shape: ring (all at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"R\"}},{\"type\":\"text\",\"text\":\"), disc (spread from "},{"type":"inlineMath","attrs":{"latex":"0\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" to \"}},{\"type\":\"text\",\"text\":\"R\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"), square loop (corners farther than \"}},{\"type\":\"text\",\"text\":\"R"}},{"type":"text","text":").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the objects based on how much mass is farthest from the axis.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q5. Pulleys with Different Radii—Rotation and Torque\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque and Rotational Acceleration\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question explores how different torques from weights on pulleys with different radii affect the direction of rotation and net torque.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r \\\\times F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Angular acceleration: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\alpha = \\\\frac{\\\\tau_{\\\\text{net}}}{I}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Right-hand rule for torque direction\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify the direction of the forces and the radii at which they act.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Use the right-hand rule to determine the direction of each torque (out of the page = counterclockwise).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the torques: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = rF\"}},{\"type\":\"text\",\"text\":\" for each weight, considering which radius and force is larger.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Relate the net torque direction to the observed angular acceleration.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q6. Spheres Rolling—Radius\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rolling Motion Without Slipping\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank the radii of rolling spheres given their angular and linear speeds.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Relationship for rolling without slipping: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v = \\\\omega R\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each sphere, calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"R = \\\\frac{v}{\\\\omega}\"}},{\"type\":\"text\",\"text\":\" using the given values.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the calculated radii to rank them from greatest to least.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Remember that all spheres have the same mass, so only \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\" matter for this ranking.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q7. Three Equal Forces Applied to a Rectangle—Net Torque Direction\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque and Rotational Equilibrium\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question examines how the direction and point of application of forces affect the net torque about different points on a rectangle.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F \\\\sin \\\\theta\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Net torque is the sum of individual torques about a point.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each force, determine the perpendicular distance from the pivot point to the line of action of the force.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Decide whether each force produces a clockwise or counterclockwise torque about the chosen point.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Sum the torques, considering their directions, to determine the net torque about each point.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the magnitudes and directions to answer each part.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q8. Spheres Rolling—Rotational Kinetic Energy\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rotational Kinetic Energy\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank the rotational kinetic energy of rolling spheres with different speeds and radii.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rotational kinetic energy: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"K_{\\\\text{rot}} = \\\\frac{1}{2} I \\\\omega^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For a hollow sphere: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I = \\\\frac{2}{3} m R^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Relationship between \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\" for rolling: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v = \\\\omega R\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each sphere, use the given \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v\"}},{\"type\":\"text\",\"text\":\" to find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"R\"}},{\"type\":\"text\",\"text\":\" if needed.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"K_{\\\\text{rot}}\"}},{\"type\":\"text\",\"text\":\" for each using the formula above.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Since mass is constant, compare \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega^2\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"R^2\"}},{\"type\":\"text\",\"text\":\" for each sphere to rank their rotational kinetic energies.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q9. Three Forces Applied to a Rectangle—Torque Direction\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque Calculation About Different Points\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to determine the direction of the torque produced by each force about various points on a rectangle.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F \\\\sin \\\\theta\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Direction: Clockwise (CW) or Counterclockwise (CCW) depending on the force's tendency to rotate the object about the point.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each force and pivot point, determine the moment arm (perpendicular distance from the point to the line of action).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Decide if the force would cause a CW or CCW rotation about the point.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"If the line of action passes through the point, the torque is zero.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Repeat for each combination as required.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q10. Fishing Rod—Weight of Two Pieces\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rotational Equilibrium and Center of Mass\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question explores how the balance point of a rod relates to the weights of its two segments if cut at that point.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rotational equilibrium: Net torque about the pivot is zero.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Identify the balancing point as the location where the net torque is zero.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Consider the torques produced by the weights of each segment about the balancing point.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recall that the farther the center of mass is from the pivot, the less weight is needed to balance the torque.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the lengths and likely positions of the centers of mass for each segment.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q11. Suspended Signs—Torque\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque Due to Hanging Weights\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank the torque exerted by signs of different masses and positions about the point where the supporting rod is attached to a building.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}},{\"type\":\"text\",\"text\":\" where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"F = mg\"}},{\"type\":\"text\",\"text\":\" (weight of the sign), \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r\"}},{\"type\":\"text\",\"text\":\" is the perpendicular distance from the pivot to the line of action of the force.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each case, identify the mass of the sign and the distance from the pivot to where the force acts.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate the perpendicular distance for each case (consider the angle if the rod is not horizontal).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply the mass (times \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"g\"}},{\"type\":\"text\",\"text\":\") by the distance to get the torque for each case.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the torques based on these products.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q12. Four Forces Acting on a Hexagon—Torque About Center\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque and Perpendicular Distance\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank the torques produced by four forces acting on a hexagon about its center.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}},{\"type\":\"text\",\"text\":\" where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r\"}},{\"type\":\"text\",\"text\":\" is the perpendicular distance from the center to the line of action of the force.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each force, determine if its line of action passes through the center (if so, torque is zero).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For the others, calculate the perpendicular distance from the center to the line of action.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply the force by this distance to get the torque for each.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the torques based on these values.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q13. Balance Beam—Motion After Release\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rotational Equilibrium and Torque\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to predict the motion of a balance beam with unequal numbers of keys at different distances from the pivot when released.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Net torque determines the direction of rotation.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate the total torque produced by the keys on each side: multiply the number of keys by their distance from the pivot.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the torques to determine which side will rotate downward.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Predict the direction of rotation based on which side has the greater torque.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q14. Rolling Objects Released from Rest—Time Down Ramp\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Conservation of Energy in Rolling Motion\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank objects by the time it takes them to roll down a ramp, considering their moments of inertia.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Conservation of energy: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"mgh = \\\\frac{1}{2}mv^2 + \\\\frac{1}{2}I\\\\omega^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Moment of inertia for solid sphere: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I = \\\\frac{2}{5}MR^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Moment of inertia for hollow sphere: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I = \\\\frac{2}{3}MR^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Moment of inertia for hoop: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I = MR^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For rolling without slipping: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v = \\\\omega R\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the energy conservation equation for each object.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Express \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\omega\"}},{\"type\":\"text\",\"text\":\" in terms of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"R\"}},{\"type\":\"text\",\"text\":\" to eliminate "},{"type":"inlineMath","attrs":{"latex":"\\\\omega"}},{"type":"text","text":".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"v\"}},{\"type\":\"text\",\"text\":\" at the bottom in terms of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"g\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"h\"}},{\"type\":\"text\",\"text\":\", and the moment of inertia coefficient.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the objects by their final speeds (faster objects reach the bottom first).\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q15. Pulley with Hanging Weights—Angular Acceleration\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Net Torque and Angular Acceleration\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to evaluate a student's reasoning about the direction of angular acceleration in a two-pulley system with different weights and radii.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Net torque determines the direction of angular acceleration.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each pulley, calculate the torque produced by the hanging weight: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r \\\\times mg\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the torques, considering the radii and weights.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Determine the direction of the net torque and thus the angular acceleration.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Assess whether the student's reasoning matches the physical situation.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q16. Tilted Pivoted Rods with Various Loads—Force to Hold Rods\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Rotational Equilibrium and Lever Arms\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank the force needed to hold rods with different masses and lever arms in equilibrium.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For equilibrium: sum of clockwise torques = sum of counterclockwise torques\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each rod, identify the mass and the distance from the pivot to where the mass is attached.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the torque balance equation: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"F_{\\\\text{applied}} \\\\times d_{\\\\text{applied}} = m \\\\times g \\\\times d_{\\\\text{mass}}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"F_{\\\\text{applied}}\"}},{\"type\":\"text\",\"text\":\" for each case and compare the values.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the cases based on the required force.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q17. Special Rod—Moment of Inertia\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Moment of Inertia and Mass Distribution\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to compare the moment of inertia of a rod with unequal mass segments about its two ends.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Moment of inertia: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I = \\\\sum m_i r_i^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Greater mass farther from the axis increases \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each axis (left and right end), calculate the distance of each mass segment from the axis.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply each mass by the square of its distance from the axis and sum for all segments.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the total moments of inertia for each axis.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Determine which configuration has more mass farther from the axis.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q18. Tilted Pivoted Rods with Various Loads—Force to Hold Rods (Shorter Rods)\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque and Rotational Equilibrium\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question compares the force needed to hold rods of different lengths but with the same mass and mass positions in equilibrium.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For equilibrium: sum of torques = 0\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each rod, identify the distances from the pivot to the mass and to the applied force.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the torque balance equation for each case.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the ratios of distances to see if the required force changes with rod length.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Determine if the force is the same or different in each case.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q19. Horizontal Pivoted Rods with Loads I—Force to Hold\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque and Rotational Equilibrium\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank the force needed to hold rods with weights at different positions in equilibrium.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For equilibrium: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"F_{\\\\text{applied}} \\\\times d_{\\\\text{applied}} = W \\\\times d_{\\\\text{weight}}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each case, identify the distances from the pivot to the applied force and to the weight.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the torque balance equation for each configuration.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Solve for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"F_{\\\\text{applied}}\"}},{\"type\":\"text\",\"text\":\" and compare the values for each case.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the cases based on the required force.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q20. Horizontal Pivoted Board with Load II—Force to Hold Board\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Effects of Changing Parameters on Torque and Equilibrium\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to predict how changes in the position of the weight or force, or the value of the weight, affect the force needed to keep a board in equilibrium.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For equilibrium: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"F_{\\\\text{applied}} \\\\times L_2 = W \\\\times L_1\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each modification, consider how it changes the torque produced by the weight or the force.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Predict whether the required force increases, decreases, or stays the same based on the new distances or weights.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For indeterminate cases, recognize when more information is needed (e.g., both distances change).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Explain your reasoning for each scenario.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q21. Hanging Weights on Fixed Disks—Torque\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Torque Produced by Hanging Weights\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank the torques produced by weights hanging from strings wrapped around disks of different diameters.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = r F\"}},{\"type\":\"text\",\"text\":\" where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"F\"}},{\"type\":\"text\",\"text\":\" is the weight (\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"mg\"}},{\"type\":\"text\",\"text\":\") and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"r\"}},{\"type\":\"text\",\"text\":\" is the radius (half the diameter).\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each disk, calculate the torque: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tau = (\\\\text{diameter}/2) \\\\times (\\\\text{hanging mass} \\\\times g)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the torques for each case.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rank the cases based on the calculated torques.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q22. Systems of Point Masses—Difficulty to Rotate\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Moment of Inertia and Rotational Dynamics\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question asks you to rank different arrangements of point masses by how difficult they are to rotate (i.e., their moment of inertia).\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Moment of inertia: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I = \\\\sum m_i r_i^2\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Greater mass farther from the axis increases \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"I\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"For each arrangement, count how many masses are far from the axis of rotation.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate or estimate the sum \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sum m_i r_i^2\"}},{\"type\":\"text\",\"text\":\" for each ca