CAIE FP2 2017 November — Question 2 7 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2017
SessionNovember
Marks7
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TopicSimple Harmonic Motion
TypeFind period from given information
DifficultyStandard +0.8 This is a standard SHM problem requiring students to set up and solve simultaneous equations using v² = ω²(a² - x²), but the algebraic manipulation is somewhat involved with non-trivial numerical values. It tests understanding of the SHM velocity formula and requires careful algebraic work to extract both amplitude and period, making it moderately challenging but still within typical Further Maths scope.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x
2 The piston in a large engine rises and falls in simple harmonic motion. When the piston is 1.6 m below its highest level, the rate of change of its height is \(\frac { 3 } { 5 } \pi\) metres per second. When the piston is 0.2 m below its highest level, the rate of change of its height is \(\frac { 1 } { 4 } \pi\) metres per second. Find the amplitude and period of the motion.
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
\((3\pi/5)^2 = \omega^2(a^2 - (a-1.6)^2)\)M1 A1 Use \(v^2 = \omega^2(a^2 - x^2)\) in each position (M1 for either)
\((\pi/4)^2 = \omega^2(a^2 - (a-0.2)^2)\); \((3\pi/5)^2(0.4a - 0.04) = (\pi/4)^2(3.2a - 2.56)\)A1 Combine to find amplitude \(a\) and \(\omega^2\) (or \(\omega\))
\(a = 2.6\ \text{m}\)M1 A1 (M1 for either)
\(\omega^2 = (\pi/4)^2\ \text{or}\ \omega = \pi/4\)A1 Find other unknown
\(T = 2\pi / (\pi/4) = 8\ \text{s}\)B1FT Find period \(T\) from \(T = 2\pi/\omega\) (\(\sqrt{}\) on \(\omega^2\) or \(\omega\))
7 
# Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $(3\pi/5)^2 = \omega^2(a^2 - (a-1.6)^2)$ | **M1 A1** | Use $v^2 = \omega^2(a^2 - x^2)$ in each position (M1 for either) |
| $(\pi/4)^2 = \omega^2(a^2 - (a-0.2)^2)$; $(3\pi/5)^2(0.4a - 0.04) = (\pi/4)^2(3.2a - 2.56)$ | **A1** | Combine to find amplitude $a$ and $\omega^2$ (or $\omega$) |
| $a = 2.6\ \text{m}$ | **M1 A1** | (M1 for either) |
| $\omega^2 = (\pi/4)^2\ \text{or}\ \omega = \pi/4$ | **A1** | Find other unknown |
| $T = 2\pi / (\pi/4) = 8\ \text{s}$ | **B1FT** | Find period $T$ from $T = 2\pi/\omega$ ($\sqrt{}$ on $\omega^2$ or $\omega$) |
| | **7** | |
2 The piston in a large engine rises and falls in simple harmonic motion. When the piston is 1.6 m below its highest level, the rate of change of its height is $\frac { 3 } { 5 } \pi$ metres per second. When the piston is 0.2 m below its highest level, the rate of change of its height is $\frac { 1 } { 4 } \pi$ metres per second. Find the amplitude and period of the motion.\\

\hfill \mbox{\textit{CAIE FP2 2017 Q2 [7]}}
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