to solve for $\\lambda$:\n$$\\lambda = \\frac{v}{f} = \\frac{1.2\\text{ m s}^{-1}}{5.0\\text{ Hz}} = 0.24\\text{ m} = 24\\text{ cm}$$"},{"type":"text","order":1,"title":"Eliminate Option D","content":"The positions $A$ and $C$ are separated by half a wavelength ($\\frac{\\lambda}{2} = 12\\text{ cm}$), which corresponds to a phase difference of $\\pi\\text{ rad}$ ($180^\\circ$). Particles separated by $\\pi\\text{ rad}$ are in **anti-phase** (completely out of phase), not in phase.\n\nTherefore, we can eliminate Option D.","highlights":[{"text":"vibrate in phase","index":3,"location":"eliminate"}]},{"type":"text","order":2,"title":"Eliminate Option C","content":"Point $B$ is located at a crest (maximum displacement), which is one-quarter of a wavelength ($6\\text{ cm}$) from $A$. \n\nIn simple harmonic motion, a particle at its maximum displacement is instantaneously at rest, meaning its velocity is **zero** (minimum), not maximum. \n\nTherefore, we can eliminate Option C.","highlights":[{"text":"maximum value","index":2,"location":"eliminate"}]},{"type":"text","order":3,"title":"Eliminate Option B","content":"Point $D$ is at a trough, which is located at a distance of $\\frac{3\\lambda}{4}$ from $A$.\n\nThe phase difference ($\\Delta \\phi$) between $A$ and $D$ is:\n$$\\Delta \\phi = \\frac{2\\pi}{\\lambda} \\times \\frac{3\\lambda}{4} = \\frac{3\\pi}{2}\\text{ rad}$$\n\nSince the phase difference is $\\frac{3\\pi}{2}\\text{ rad}$ (or $270^\\circ$), Option B is incorrect and can be eliminated.","highlights":[{"text":"is \\frac{\\pi}{2}\\text{ rad}","index":1,"location":"eliminate"}]},{"type":"text","order":4,"title":"Confirm the correct answer","content":"Since points $A$ and $C$ are separated by exactly half a wavelength:\n$$\\text{Distance } AC = \\frac{\\lambda}{2} = \\frac{24\\text{ cm}}{2} = 12\\text{ cm}$$\n\nThis perfectly confirms that **Option A** is the correct statement.","highlights":[{"text":"The distance between $A$ and $C$ is $12\\text{ cm}$.","index":0,"location":"option"}]}]}],"generatedAt":"2026-06-10T02:25:53.050Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"af5f8481-b36a-4bc7-b757-06c0df1b05ef","revisionId":"2f08d7ae-01d5-4328-af6f-9515a27ddab3","questionNumber":null,"questionType":"LA","specification":"","difficulty":6,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"dd73edb4-5242-43a8-b226-8b9e5da4bf88","label":null,"content":"In an experimental acoustics laboratory, a stationary wave is set up on a high-tension carbon-fiber monofilament (String A) of length \$L_A\$ and diameter \$D_A\$, under a tension \$T_A\$.\n\nA second monofilament (String B) is made from the same carbon-fiber material but has a length \$L_B = 1.6 L_A\$ and is subjected to a tension \$T_B = 1.44 T_A\$.\n\nDetermine the ratio of the diameter of String B to String A, \$\\frac{D_B}{D_A}\$, required for String B to vibrate with the same fundamental (first-harmonic) frequency as String A.","markscheme":"- Recalls the formula for the fundamental frequency of a stretched string, $f = \\frac{1}{2L} \\sqrt{\\frac{T}{\\mu}}$, and relates mass per unit length to density and diameter: $\\mu = \\rho A = \\rho \\pi \\frac{D^2}{4}$ M1\n- Substitutes $\\mu$ to express frequency in terms of diameter: $f = \\frac{1}{LD} \\sqrt{\\frac{T}{\\rho \\pi}}$ (or states that $f \\propto \\frac{\\sqrt{T}}{LD}$ for constant density) M1\n- Equates the frequencies ($f_A = f_B$) to set up the ratio equation: $\\frac{D_B}{D_A} = \\frac{L_A}{L_B} \\sqrt{\\frac{T_B}{T_A}}$ M1\n- Substitutes the proportional values $L_B = 1.6 L_A$ and $T_B = 1.44 T_A$ into the ratio equation: $\\frac{D_B}{D_A} = \\frac{1}{1.6} \\sqrt{1.44}$ M1\n- Evaluates the final ratio to obtain $\\frac{D_B}{D_A} = 0.75$ (or $\\frac{3}{4}$) A1","marks":5,"order":0,"markingRubric":null,"explanation":{"methods":[{"id":"proportionality","label":"Proportionality Reasoning","steps":[{"type":"text","order":0,"tit

Progressive and stationary waves

Question 29

Progressive and stationary waves Questions

Progressive and stationary waves

Question 29

Progressive and stationary waves Questions