CAIE FP2 2009 June — Question 2 7 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2009
SessionJune
Marks7
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Mark schemeDownload PDF ↗
TopicSimple Harmonic Motion
TypeMaximum speed in SHM
DifficultyModerate -0.5 This is a straightforward application of standard SHM formulas. Finding maximum speed requires v_max = ωa (where ω = 2π/T and amplitude a is given), then the second part uses v = ω√(a² - x²). Both are direct substitutions into memorized formulas with no conceptual difficulty or problem-solving required, making it slightly easier than average.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x
2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm . It takes 2.25 s to perform 100 complete oscillations. Find, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), the maximum speed of the tip of the needle. Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is \(2.50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
Question 2:
AnswerMarks Guidance
Working/AnswerMark Guidance
\(\omega = 2\pi/0.0225\) Find frequency \(\omega\) using \(T = 2\pi/\omega\)
\([= 800\pi/9 = 279.25]\)M1 A1
\(v_{max} = 0.0105\,\omega = 2.93_{[2]}\)M1 A1; A1 Find \(v_{max}\) using \(v_{max} = a\omega\). Part mark: 5
\(v = \omega\sqrt{(0.0105^2 - 0.0055^2)}\) Find \(v\) using \(v^2 = \omega^2(a^2 - x^2)\)
\([t = 0.00197]\), \(v = 2.50\) A.G.M1 A1 *or* \(\omega t = \sin^{-1}(x/a) = [0.5513]\), \(v = a\omega\cos\omega t\). Part mark: 2. Total: [7] 
## Question 2:

| Working/Answer | Mark | Guidance |
|---|---|---|
| $\omega = 2\pi/0.0225$ | | Find frequency $\omega$ using $T = 2\pi/\omega$ |
| $[= 800\pi/9 = 279.25]$ | M1 A1 | |
| $v_{max} = 0.0105\,\omega = 2.93_{[2]}$ | M1 A1; A1 | Find $v_{max}$ using $v_{max} = a\omega$. Part mark: 5 |
| $v = \omega\sqrt{(0.0105^2 - 0.0055^2)}$ | | Find $v$ using $v^2 = \omega^2(a^2 - x^2)$ |
| $[t = 0.00197]$, $v = 2.50$ **A.G.** | M1 A1 | *or* $\omega t = \sin^{-1}(x/a) = [0.5513]$, $v = a\omega\cos\omega t$. Part mark: 2. Total: **[7]** |

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2 The tip of a sewing-machine needle oscillates vertically in simple harmonic motion through a distance of 2.10 cm . It takes 2.25 s to perform 100 complete oscillations. Find, in $\mathrm { m } \mathrm { s } ^ { - 1 }$, the maximum speed of the tip of the needle.

Show that the speed of the tip when it is at a distance of 0.5 cm from a position of instantaneous rest is $2.50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 3 significant figures.

\hfill \mbox{\textit{CAIE FP2 2009 Q2 [7]}}
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