scheme":"- Uses $x = A \\cos(2\\pi f t)$ or $x = A \\cos(\\omega t)$ with $\\omega = 2\\pi f = 3\\pi\\text{ rad s}^{-1}$ M1\n- Substitutes $A = 2.5\\text{ cm}$, $f = 1.5\\text{ Hz}$ and $t = 0.40\\text{ s}$ to get $x = 2.5 \\cos(1.2\\pi)$ M1\n- Obtains $x = -2.0\\text{ cm}$ (accept $-2.02\\text{ cm}$ or $-2.0\\text{ cm}$) A1","marks":3,"order":1,"markingRubric":null,"explanation":{"methods":[{"id":"cosine_shm","label":"Cosine model for SHM","steps":[{"type":"text","order":0,"title":"Choose the model","content":"Because the hydrometer starts at its **maximum displacement**, use the cosine form of free simple harmonic motion:\n\n$$x=A\\cos(2\\pi ft)$$\n\nCosine is appropriate because $\\cos 0=1$, so at release $x=A$. Choosing this model, or equivalently $x=A\\cos(\\omega t)$ with $\\omega=2\\pi f=3\\pi\\ \\text{rad s}^{-1}$, earns the first **M1** mark.","highlights":[{"text":"released from its maximum displacement of \$2.5\\text{ cm}\$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=\\htmlData{anchor=model}{\\color{@sky}{A\\cos(2\\pi f t)}}$"},{"latex":"$$x=\\htmlData{anchor=substitution}{\\color{@sky}{2.5\\cos(2\\pi\\times1.5\\times0.40)}}$"},{"latex":"$$x=2.5\\cos(1.2\\pi)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"model"},{"color":"sky","style":"text-color","anchor":"substitution"}],"connections":[{"to":"substitution","from":"model","color":"sky","label":"substitute"}]},"order":1,"title":"Substitute the data","description":"Keep the amplitude in centimetres because the required displacement is in centimetres. This complete substitution earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=2.5\\cos(1.2\\pi)$"},{"latex":"$$x=\\htmlData{anchor=unrounded}{-2.0225\\ldots\\ \\text{cm}}$"},{"latex":"$$x=\\htmlData{anchor=rounded}{-2.0\\ \\text{cm}}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"rounded"}],"connections":[{"to":"rounded","from":"unrounded","color":"green","label":"round suitably"}]},"order":2,"title":"Evaluate the displacement","description":"Evaluate the cosine in radians because the phase was calculated using 2πft. The result −2.0 cm earns the A1 mark. The negative sign means the hydrometer is on the opposite side of equilibrium from its initial displacement."}]}],"generatedAt":"2026-07-20T16:09:44.300Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"f2af2dbb-2ca8-4feb-99d0-a83f4fac4666","label":"c","content":"Calculate the maximum speed of the oscillating hydrometer (in \$\\text{m s}^{-1}\$) to 2 significant figures.","markscheme":"- Uses $v_{\\max} = 2\\pi f A$ or $v_{\\max} = \\omega A$ M1\n- Substitutes $f = 1.5\\text{ Hz}$ and $A = 0.025\\text{ m}$ to get $v_{\\max} = 2\\pi \\times 1.5 \\times 0.025$ M1\n- Obtains $v_{\\max} = 0.24\\text{ m s}^{-1}$ to 2 s.f. A1","marks":3,"order":2,"markingRubric":null,"explanation":{"methods":[{"id":"maximum_shm_speed","label":"Use the SHM speed formula","steps":[{"type":"text","order":0,"title":"Choose the formula","content":"For simple harmonic motion, the greatest speed occurs as the hydrometer passes through its equilibrium position. Use\n\n$$v_{\\max}=\\omega A=2\\pi fA$$\n\nwhere $f$ is frequency and $A$ is amplitude. Choosing either form earns the first **M1** mark.","highlights":[{"text":"maximum speed","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$v_{\\max}=\\htmlData{anchor=formula}{\\color{@sky}{2\\pi fA}}$"},{"latex":"$$v_{\\max}=\\htmlData{anchor=substitution}{\\color{@sky}{2\\pi\\times1.5\\times0.025}}$"},{"latex":"$$v_{\\max}=\\htmlData{anchor=unrounded}{\\color{@orange}{0.235619449\\ldots\\ \\mathrm{m\\,s^{-1}}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substitution"},{"color":"orange","style":"text-color","anchor":"unrounded"}],"connections":[{"to":"substitution","from":"formula","color":"sky","style":"solid"}]},"order":1,"title":"Substitute the value

Oscillations

Question 3

Oscillations Questions

Oscillations

Question 3

Oscillations Questions