CAIE FP2 2016 June — Question 3 8 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2016
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSimple Harmonic Motion
TypeFind period from given information
DifficultyChallenging +1.2 This is a Further Maths SHM question requiring understanding of the relationship between speed and position (v = ω√(a² - x²)), and solving for when v ≤ v_max/2. It involves setting up an equation from the time constraint, using symmetry, and inverse trig, but follows a standard SHM framework with clear given information. More challenging than routine C4/FP1 questions but not requiring exceptional insight.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x4.10g Damped oscillations: model and interpret
3 A particle \(P\) is performing simple harmonic motion with amplitude 0.25 m . During each complete oscillation, \(P\) moves with a speed that is less than or equal to half of its maximum speed for \(\frac { 4 } { 3 }\) seconds. Find the period of the motion and the maximum speed of \(P\).
Question 3:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(x = a\cos\omega t\), \(v = (\pm)\,a\omega\sin\omega t\)M1 EITHER: Find time during which speed is low
\(\frac{1}{2}a\omega = a\omega\sin\omega t\)M1 A1 (working may be implied)
\(t_L = \pi/6\omega\) or \(T/12\)M1 A1
\(4t_L = 4/3\), \(\omega = \pi/2\) or \(T = 4\) [s]B1\(\checkmark\) Find \(\omega\) (implied?) or \(T\) from given time
\(T = 2\pi/\omega = 4\) [s] or \(\omega = \pi/2\)M1 A1 Find other value (\(\checkmark\) on first)
\(v_{\max} = 0.25\omega = \pi/8\) or \(0.393\) [m] Find maximum speed \(v_{\max}\)
OR: \(x = a\sin\omega t\), \(v = a\omega\cos\omega t\)(M1) Find time during which speed is high
\(\frac{1}{2}a\omega = a\omega\cos\omega t\)(M1 A1) (working may be implied)
\(t_H = \pi/3\omega\) or \(T/6\)(M1 A1)
\(2\pi/\omega - 4t_L = 4/3\), \(\omega = \pi/2\) or \(T = 4\) [s]B1\(\checkmark\)
\(T = 2\pi/\omega = 4\) [s] or \(\omega = \pi/2\)(M1 A1) Total: [8]
OR: \(\omega^2(a^2 - x^2) = \frac{1}{4}\omega^2 a^2\)(B1) Find \(x\) when speed is \(\frac{1}{2}v_{\max}\) (AEF)
\(x^2 = \frac{3}{4}a^2\), \([\pm]\,x = \sqrt{3}a/2\) or \(\sqrt{3}/8\)(M1 A1)
\(\sqrt{3}a/2 = a\cos(\omega t)\), \(\omega t_L = \pi/6\)
\(\sqrt{3}a/2 = a\sin(\omega t)\), \(\omega t_H = \pi/3\) Total: [8] 
## Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = a\cos\omega t$, $v = (\pm)\,a\omega\sin\omega t$ | M1 | EITHER: Find time during which speed is low |
| $\frac{1}{2}a\omega = a\omega\sin\omega t$ | M1 A1 | (working may be implied) |
| $t_L = \pi/6\omega$ or $T/12$ | M1 A1 | |
| $4t_L = 4/3$, $\omega = \pi/2$ or $T = 4$ [s] | B1$\checkmark$ | Find $\omega$ (implied?) or $T$ from given time |
| $T = 2\pi/\omega = 4$ [s] or $\omega = \pi/2$ | M1 A1 | Find other value ($\checkmark$ on first) |
| $v_{\max} = 0.25\omega = \pi/8$ or $0.393$ [m] | | Find maximum speed $v_{\max}$ |
| **OR:** $x = a\sin\omega t$, $v = a\omega\cos\omega t$ | (M1) | Find time during which speed is high |
| $\frac{1}{2}a\omega = a\omega\cos\omega t$ | (M1 A1) | (working may be implied) |
| $t_H = \pi/3\omega$ or $T/6$ | (M1 A1) | |
| $2\pi/\omega - 4t_L = 4/3$, $\omega = \pi/2$ or $T = 4$ [s] | B1$\checkmark$ | |
| $T = 2\pi/\omega = 4$ [s] or $\omega = \pi/2$ | (M1 A1) | Total: [8] |
| **OR:** $\omega^2(a^2 - x^2) = \frac{1}{4}\omega^2 a^2$ | (B1) | Find $x$ when speed is $\frac{1}{2}v_{\max}$ (AEF) |
| $x^2 = \frac{3}{4}a^2$, $[\pm]\,x = \sqrt{3}a/2$ or $\sqrt{3}/8$ | (M1 A1) | |
| $\sqrt{3}a/2 = a\cos(\omega t)$, $\omega t_L = \pi/6$ | | |
| $\sqrt{3}a/2 = a\sin(\omega t)$, $\omega t_H = \pi/3$ | | Total: [8] |

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3 A particle $P$ is performing simple harmonic motion with amplitude 0.25 m . During each complete oscillation, $P$ moves with a speed that is less than or equal to half of its maximum speed for $\frac { 4 } { 3 }$ seconds. Find the period of the motion and the maximum speed of $P$.

\hfill \mbox{\textit{CAIE FP2 2016 Q3 [8]}}
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Q1 5 Q2 8 Q3 8 Q4 10 Q5 12 Q6 5 Q7 8 Q8 9 Q9 10 Q10 11 Q11 EITHER Q11 OR