{ "format": "recallweave-deck/1", "title": "Coupled motion: two patterns inside one system", "attribution": "Original AI-assisted questions, distractors, explanations, examples and diagrams for RecallWeave. Mathematical background: MIT OpenCourseWare 8.03SC, Chapter 3, Normal Modes (Fall 2016), https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/d96262076cff658e551107bbb7c4b14c_MIT8_03SCF16_Text_Ch3.pdf. This course derives an ideal two-mass, three-spring model using half-sum normal coordinates; no source exercises, passages or figures are copied.", "license": "Original question wording, explanations, worked examples and diagrams in this course: CC0-1.0. Referenced materials retain their own licenses. Other repository content is unchanged.", "concepts": [ "restoring-and-coupling", "normal-coordinates", "initial-motion", "energy-and-limits" ], "items": [ { "id": "modes-force-balance", "concept": "restoring-and-coupling", "prerequisites": [], "prompt": "Two equal masses move horizontally. Each has a wall spring k, and a spring c connects the masses. Displacements x1 and x2 are measured rightward from equilibrium. With k = 10 N/m, c = 5 N/m, x1 = 0.10 m and x2 = 0.30 m, what is the net force on mass 1?", "options": [ "−2 N", "0 N", "+1 N", "+2 N" ], "answer": 1, "explanation": "The wall force is −k x1 = −1 N. The coupling force on mass 1 is c(x2 − x1) = +1 N. They cancel at this instant. Zero instantaneous force means zero acceleration then; it does not imply zero displacement, zero velocity, or permanent equilibrium.", "transfer": "Keep x1 at 0.10 m but change x2 to 0.10 m. Calculate both contributions and explain the sign of the new force." }, { "id": "modes-common-displacement", "concept": "restoring-and-coupling", "prerequisites": [], "prompt": "In the same ideal two-mass model, both displacements equal +0.20 m. The wall stiffness is k = 8 N/m and the coupling stiffness is positive. Which statement describes the coupling spring at this instant?", "options": [ "Its extension is 0.40 m and it pushes both masses right.", "Its extension is 0.20 m and it stops both masses.", "Its extension from equilibrium is zero, so its force is zero.", "Its force is unknown until the masses are specified." ], "answer": 2, "explanation": "The change in the separation of the masses is x2 − x1 = 0. Equal displacements leave the coupling spring's extension unchanged. Its force is zero, while each wall spring exerts −8×0.20 = −1.60 N. Equal motion does not remove the wall forces.", "transfer": "In the in-phase preset, follow a quarter period. Explain why the coupling contribution remains zero throughout that pure mode." }, { "id": "modes-opposite-acceleration", "concept": "restoring-and-coupling", "prerequisites": [], "prompt": "Let m = 2 kg for each mass, k = 8 N/m and c = 6 N/m. At one instant x1 = +0.25 m and x2 = −0.25 m. What is the acceleration of mass 1?", "options": [ "−2.5 m/s²", "−1.0 m/s²", "+2.5 m/s²", "−5.0 m/s²" ], "answer": 0, "explanation": "F1 = −k x1 + c(x2 − x1) = −2 −3 = −5 N. Dividing by m = 2 kg gives a1 = −2.5 m/s². Opposite displacements stretch or compress the central spring twice as much as either individual displacement.", "transfer": "Calculate the acceleration of mass 2, then explain the symmetry without starting a second full calculation." }, { "id": "modes-total-force", "concept": "restoring-and-coupling", "prerequisites": [], "prompt": "For two equal masses with wall stiffness k and coupling stiffness c, add the two equations F1 = −k x1 + c(x2 − x1) and F2 = −k x2 + c(x1 − x2). What is the total force on the two masses?", "options": [ "−(k + 2c)(x1 + x2)", "0 for every configuration", "c(x2 − x1)", "−k(x1 + x2)" ], "answer": 3, "explanation": "The equal and opposite coupling forces cancel in the sum. The two wall forces remain, giving −k(x1 + x2). The masses are not an isolated system for momentum because the fixed walls exert external forces.", "transfer": "If x1 + x2 = 0 at one instant but v1 + v2 is nonzero, does that guarantee the sum of positions remains zero? Explain." }, { "id": "modes-half-sum", "concept": "normal-coordinates", "prerequisites": [ "restoring-and-coupling" ], "prompt": "Use q+ = (x1 + x2)/2 and q− = (x1 − x2)/2. If x1 = +0.60 m and x2 = −0.20 m, what are the two normal coordinates?", "options": [ "q+ = 0.40 m; q− = 0.80 m", "q+ = 0.20 m; q− = 0.40 m", "q+ = 0.40 m; q− = 0.20 m", "q+ = −0.20 m; q− = 0.60 m" ], "answer": 1, "explanation": "The half-sum is (0.60 −0.20)/2 = 0.20 m; the half-difference is (0.60 +0.20)/2 = 0.40 m. Reconstruction gives x1 = q+ + q− = 0.60 m and x2 = q+ − q− = −0.20 m. The factor of one half is part of this course's coordinate convention.", "transfer": "Reconstruct the physical positions for q+ = −0.10 m and q− = +0.30 m." }, { "id": "modes-frequencies", "concept": "normal-coordinates", "prerequisites": [ "restoring-and-coupling" ], "prompt": "For two equal masses m = 2 kg, wall stiffness k = 8 N/m and coupling stiffness c = 6 N/m, which pair gives the in-phase and opposite-motion angular frequencies, respectively?", "options": [ "2 rad/s and √7 rad/s", "√8 rad/s and √20 rad/s", "2 rad/s and √10 rad/s", "4 rad/s and 10 rad/s" ], "answer": 2, "explanation": "The half-sum obeys q+'' = −(k/m)q+, so ω+ = √(8/2) = 2 rad/s. The half-difference obeys q−'' = −((k+2c)/m)q−, so ω− = √(20/2) = √10 rad/s. These are angular frequencies; divide by 2π to obtain cycles per second.", "transfer": "Increase only c. Which frequency changes, and what physical deformation accounts for that change?" }, { "id": "modes-stiffer-coupling", "concept": "normal-coordinates", "prerequisites": [ "restoring-and-coupling" ], "prompt": "Keep m and k fixed and positive, then increase the nonnegative coupling stiffness c. What happens to the two angular frequencies?", "options": [ "Both decrease.", "The in-phase frequency stays fixed; the opposite-motion frequency increases.", "Both increase by the same amount.", "The opposite-motion frequency stays fixed; the in-phase frequency increases." ], "answer": 1, "explanation": "In the in-phase pattern, the coupling spring has no changing extension, so ω+ = √(k/m) is independent of c. Opposite motion changes the separation; ω− = √((k+2c)/m) increases with c. The increase is not generally linear in c.", "transfer": "Set the coupling to zero and then restore it. Describe what changes in the two mode traces without changing the initial state." }, { "id": "modes-degenerate-basis", "concept": "normal-coordinates", "prerequisites": [ "restoring-and-coupling" ], "prompt": "Set c = 0 while m and k remain positive. Which statement is correct?", "options": [ "The two masses are uncoupled and both natural angular frequencies equal √(k/m); the chosen normal-coordinate basis is not unique.", "Both natural frequencies become zero.", "Only the in-phase pattern remains possible.", "The total mechanical energy must become zero." ], "answer": 0, "explanation": "Removing the coupling leaves two independent, identical wall-spring oscillators. Their frequencies coincide. The half-sum and half-difference remain a valid basis, but other independent combinations also work at this repeated frequency. Frequency degeneracy does not erase the possible motions or their energy.", "transfer": "Can one mass oscillate while the other stays at equilibrium when c = 0? Give the required initial state." }, { "id": "modes-position-not-enough", "concept": "initial-motion", "prerequisites": [ "normal-coordinates" ], "prompt": "At t = 0, x1 = x2 = 0.40 m, but v1 = +0.20 m/s and v2 = −0.20 m/s. The coupling c is positive. Is this a pure in-phase motion?", "options": [ "Yes, equal starting positions are sufficient.", "Yes, because the total initial velocity is zero.", "No, because q− starts at 0.40 m.", "No, because q− starts at zero but its initial velocity is 0.20 m/s." ], "answer": 3, "explanation": "A pure in-phase solution requires both q−(0) = 0 and q−'(0) = 0. Equal positions give the first condition, but (v1 − v2)/2 = 0.20 m/s violates the second. The opposite-motion coordinate immediately begins to change.", "transfer": "Give one nonzero initial state that excites only the in-phase mode, including both positions and velocities." }, { "id": "modes-pure-opposite", "concept": "initial-motion", "prerequisites": [ "normal-coordinates" ], "prompt": "With positive coupling, which initial state excites only the opposite-motion mode?", "options": [ "x1 = x2 = 0.30 m; v1 = v2 = 0.40 m/s", "x1 = 0.30 m, x2 = −0.30 m; v1 = 0.40 m/s, v2 = −0.40 m/s", "x1 = 0.30 m, x2 = −0.30 m; v1 = v2 = 0.40 m/s", "x1 = x2 = 0; v1 = 0.40 m/s, v2 = 0" ], "answer": 1, "explanation": "The in-phase coordinate must have both initial value and initial velocity zero. Opposite positions make (x1+x2)/2 = 0, and opposite velocities make (v1+v2)/2 = 0. Opposite positions alone are insufficient if the velocities contain an in-phase component.", "transfer": "Swap the labels of the two masses. Which normal coordinate changes sign, and which mode energy changes?" }, { "id": "modes-quarter-period", "concept": "initial-motion", "prerequisites": [ "normal-coordinates" ], "prompt": "A pure in-phase motion has q+(0) = 0.20 m, q+'(0) = 0 and ω+ = 2 rad/s. The opposite-motion coordinate is identically zero. At t = π/4 s, what are the physical positions and velocities?", "options": [ "Both positions are zero; both velocities are −0.40 m/s.", "Both positions are +0.20 m; both velocities are zero.", "The positions are equal and opposite; the velocities are zero.", "Both positions are zero; both velocities are +0.40 m/s." ], "answer": 0, "explanation": "This time is a quarter period because ω+t = π/2. Thus q+ = 0.20 cos(π/2) = 0 and q+' = −0.20×2 sin(π/2) = −0.40 m/s. With q− = 0, both physical positions equal q+ and both velocities equal q+'.", "transfer": "Predict the same four values at t = π/2 s before using the time control." }, { "id": "modes-uncoupled-localized", "concept": "initial-motion", "prerequisites": [ "normal-coordinates" ], "prompt": "With c = 0, start mass 1 at x1 = 0.50 m with v1 = 0. Start mass 2 at x2 = 0 with v2 = 0. What happens to mass 2 in this ideal model?", "options": [ "It gradually takes all of mass 1's energy.", "It begins to move after one quarter cycle.", "It remains at equilibrium, even though both chosen normal coordinates are excited.", "It must move because the normal frequencies are equal." ], "answer": 2, "explanation": "Without coupling, mass 2 satisfies its own oscillator equation with zero initial position and velocity, so it stays at zero. The half-sum and half-difference initially both equal 0.25 m and have the same frequency. They cancel exactly in x2 = q+ − q−. A decomposition can contain two nonzero coordinates while one physical component remains zero.", "transfer": "Restore a positive coupling without changing that initial state. Calculate mass 2's initial acceleration." }, { "id": "modes-three-spring-energy", "concept": "energy-and-limits", "prerequisites": [ "normal-coordinates", "initial-motion" ], "prompt": "At one instant m = 1 kg, k = 4 N/m, c = 6 N/m, x1 = +0.50 m, x2 = −0.50 m, and both velocities are zero. What is the total mechanical energy?", "options": [ "1 J", "2 J", "3 J", "4 J" ], "answer": 3, "explanation": "Kinetic energy is zero. The wall springs store ½k(x1²+x2²) = 1 J. The coupling spring stores ½c(x2−x1)² = 3 J, since its extension has magnitude 1 m. The total is 4 J. Omitting the central spring loses most of this state's energy.", "transfer": "Keep the positions fixed but set both velocities to +1 m/s. How much energy is added?" }, { "id": "modes-modal-normalization", "concept": "energy-and-limits", "prerequisites": [ "normal-coordinates", "initial-motion" ], "prompt": "In the half-sum convention q+ = (x1+x2)/2, let m = 2 kg, k = 8 N/m, q+ = 0.10 m and q+' = 0.20 m/s. What is the energy in the in-phase mode?", "options": [ "0.08 J", "0.16 J", "0.24 J", "0.32 J" ], "answer": 1, "explanation": "For this convention E+ = m(q+')² + k(q+)² = 2×0.20² + 8×0.10² = 0.16 J. The coordinate moves both physical masses, so its effective modal mass is 2m. Adding an extra factor of one half would mix this convention with an orthonormal coordinate normalization.", "transfer": "Derive the opposite-mode expression and identify where the additional 2c stiffness enters." }, { "id": "modes-energy-conservation", "concept": "energy-and-limits", "prerequisites": [ "normal-coordinates", "initial-motion" ], "prompt": "A mixed initial state excites both normal modes. The springs are ideal, the walls fixed, and there is no damping or drive. Which energy statement is generally valid?", "options": [ "Each individual mass keeps a constant kinetic energy.", "The coupling spring keeps a constant potential energy.", "Total mechanical energy and each modal energy remain constant, while the physical kinetic and spring energies can change.", "All the energy must transfer completely from one physical mass to the other at a regular interval." ], "answer": 2, "explanation": "The two normal coordinates obey independent conservative oscillator equations, so E+ and E− are separately constant, and their sum is the conserved total. Physical kinetic, wall-spring and coupling-spring energies vary as the modes combine. Complete transfer between physical masses is not a general consequence of a visible amplitude envelope; the coupling spring and velocity terms must be accounted for.", "transfer": "Choose the localized preset. Compare an instant of small x1 with v1 and the coupling-spring energy. Why is small displacement alone insufficient to show zero energy?" }, { "id": "modes-rest-and-frequency", "concept": "energy-and-limits", "prerequisites": [ "normal-coordinates", "initial-motion" ], "prompt": "All initial displacements and velocities are zero, while m and k are positive and c is nonnegative. What should the ideal lab show?", "options": [ "Rest at every time and zero energies, with the two natural frequencies still determined by m, k and c.", "Undefined natural frequencies because no motion was excited.", "Spontaneous oscillation after a small waiting time.", "Nonzero coupling energy whenever c is positive." ], "answer": 0, "explanation": "The zero initial state stays zero in these homogeneous linear equations. Every kinetic and spring energy is zero. The natural frequencies describe how the same system would respond to an excited initial state; they do not require the current state to be moving.", "transfer": "Keep the initial state at rest and double both masses. Which displayed quantities change, and which remain zero?" } ] }