CAIE FP2 2010 June — Question 1 5 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2010
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicSimple Harmonic Motion
TypeMaximum acceleration in SHM
DifficultyStandard +0.3 This is a straightforward SHM problem requiring standard formulas (ω = 2π/T, amplitude = 0.3m, F_max = mω²a) with simple arithmetic. It's slightly above average difficulty due to being Further Maths content and requiring correct identification of amplitude from total distance, but involves no problem-solving insight or complex manipulation.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x
1 A particle \(P\), of mass 0.2 kg , moves in simple harmonic motion along a straight line under the action of a resultant force of magnitude \(F \mathrm {~N}\). The distance between the end-points of the motion is 0.6 m , and the period of the motion is 0.5 s . Find the greatest value of \(F\) during the motion.
Question 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\omega = 2\pi / 0.5 = [4\pi = 12.57]\)B1 Find \(\omega\) or \(\omega^2\) from \(2\pi/T\)
\(F = 0.2\, d^2x/dt^2\)M1 Relate \(F\) to acceleration
\(d^2x/dt^2 = [-]\omega^2 x\)M1 Relate acceleration to \(\omega\) and \(x\)
Maximum when \(x = [\pm]\, 0.3\)M1 State or use value of \(x\) giving max of \(F\)
\(0.2(4\pi)^2 \cdot 0.3 = 0.96\pi^2\) or \(9.47\)A1 Evaluate maximum \(F_{max}\)
Total: 5 marks
## Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\omega = 2\pi / 0.5 = [4\pi = 12.57]$ | B1 | Find $\omega$ or $\omega^2$ from $2\pi/T$ |
| $F = 0.2\, d^2x/dt^2$ | M1 | Relate $F$ to acceleration |
| $d^2x/dt^2 = [-]\omega^2 x$ | M1 | Relate acceleration to $\omega$ and $x$ |
| Maximum when $x = [\pm]\, 0.3$ | M1 | State or use value of $x$ giving max of $F$ |
| $0.2(4\pi)^2 \cdot 0.3 = 0.96\pi^2$ or $9.47$ | A1 | Evaluate maximum $F_{max}$ |

**Total: 5 marks**

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1 A particle $P$, of mass 0.2 kg , moves in simple harmonic motion along a straight line under the action of a resultant force of magnitude $F \mathrm {~N}$. The distance between the end-points of the motion is 0.6 m , and the period of the motion is 0.5 s . Find the greatest value of $F$ during the motion.

\hfill \mbox{\textit{CAIE FP2 2010 Q1 [5]}}
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Q1 5 Q2 7 Q3 9 Q4 9 Q5 13 Q6 5 Q7 7 Q8 9 Q9 9 Q10 13 Q11 EITHER Q11 OR