CAIE FP2 2014 June — Question 2 8 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2014
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSimple Harmonic Motion
TypeTime to travel between positions
DifficultyChallenging +1.2 This is a two-part SHM question requiring application of energy conservation and kinematic equations. While it involves multiple steps (setting up KE ratio, solving for amplitude, then using SHM displacement formula with integration), the techniques are standard for Further Maths students. The energy approach and time calculation are well-practiced methods, making this moderately above average but not requiring novel insight.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x6.05a Angular velocity: definitions
2 The point \(O\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(O A = 0.5 \mathrm {~m}\) and \(O B = 0.75 \mathrm {~m}\), with \(A\) between \(O\) and \(B\). A particle \(P\) of mass \(m\) oscillates on \(l\) in simple harmonic motion with centre \(O\). The ratio of the kinetic energy of \(P\) when it is at \(A\) to its kinetic energy when it is at \(B\) is \(12 : 11\). Find the amplitude of the motion. Given that the greatest speed of \(P\) is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time taken by \(P\) to travel directly from \(A\) to \(B\).
Question 2:
Finding \(v^2\) at A and B:
AnswerMarks Guidance
\(v_A^2 = \omega^2(a^2 - 0.5^2)\) and \(v_B^2 = \omega^2(a^2 - 0.75^2)\)B1 Find \(v^2\) at both \(A\) and \(B\)
Finding amplitude \(a\):
AnswerMarks Guidance
\(\frac{1}{2}mv_A^2 = \frac{12}{11} \cdot \frac{1}{2}mv_B^2\)
\(11(a^2 - 0.5^2) = 12(a^2 - 0.75^2)\)
\(a^2 = \frac{1}{4}(27-11) = 4\), \(a = 2\)M1 A1 Find amplitude \(a\) m from given K.E. ratio
Finding \(\omega\):
AnswerMarks Guidance
\(0.6 = 2\omega\), \(\omega = 0.3\)B1 Find \(\omega\) from \(v_{\max} = a\omega\)
Finding time at A:
AnswerMarks Guidance
\(\omega^{-1}\sin^{-1}(0.5/2)\) or \(\omega^{-1}\cos^{-1}(0.5/2)\)M1 A1\(\checkmark\) Find time (\(\checkmark\) on \(a\)) at \(A\)
Finding time at B:
AnswerMarks Guidance
\(\omega^{-1}\sin^{-1}(0.75/2)\) or \(\omega^{-1}\cos^{-1}(0.75/2)\) or at \(B\)
Combining to find time A to B:
AnswerMarks
\(\omega^{-1}\sin^{-1}(0.75/2) - \omega^{-1}\sin^{-1}(0.5/2)\)M1
or \(\omega^{-1}\cos^{-1}(0.5/2) - \omega^{-1}\cos^{-1}(0.75/2)\)
Evaluating:
AnswerMarks Guidance
\(= \omega^{-1}(0.3844 - 0.2527)\) or \(\omega^{-1}(1.318 - 1.186)\)
\(= 1.2813 - 0.8423\)
\(4.3937 - 3.9547 = 0.439\) [s]A1 Evaluate to 3 d.p. 
# Question 2:

**Finding $v^2$ at A and B:**
| $v_A^2 = \omega^2(a^2 - 0.5^2)$ and $v_B^2 = \omega^2(a^2 - 0.75^2)$ | B1 | Find $v^2$ at both $A$ and $B$ |

**Finding amplitude $a$:**
| $\frac{1}{2}mv_A^2 = \frac{12}{11} \cdot \frac{1}{2}mv_B^2$ | | |
| $11(a^2 - 0.5^2) = 12(a^2 - 0.75^2)$ | | |
| $a^2 = \frac{1}{4}(27-11) = 4$, $a = 2$ | M1 A1 | Find amplitude $a$ m from given K.E. ratio |

**Finding $\omega$:**
| $0.6 = 2\omega$, $\omega = 0.3$ | B1 | Find $\omega$ from $v_{\max} = a\omega$ |

**Finding time at A:**
| $\omega^{-1}\sin^{-1}(0.5/2)$ or $\omega^{-1}\cos^{-1}(0.5/2)$ | M1 A1$\checkmark$ | Find time ($\checkmark$ on $a$) at $A$ |

**Finding time at B:**
| $\omega^{-1}\sin^{-1}(0.75/2)$ or $\omega^{-1}\cos^{-1}(0.75/2)$ | | or at $B$ |

**Combining to find time A to B:**
| $\omega^{-1}\sin^{-1}(0.75/2) - \omega^{-1}\sin^{-1}(0.5/2)$ | M1 | |
| or $\omega^{-1}\cos^{-1}(0.5/2) - \omega^{-1}\cos^{-1}(0.75/2)$ | | |

**Evaluating:**
| $= \omega^{-1}(0.3844 - 0.2527)$ or $\omega^{-1}(1.318 - 1.186)$ | | |
| $= 1.2813 - 0.8423$ | | |
| $4.3937 - 3.9547 = 0.439$ [s] | A1 | Evaluate to 3 d.p. |

---
2 The point $O$ is on the fixed line $l$. Points $A$ and $B$ on $l$ are such that $O A = 0.5 \mathrm {~m}$ and $O B = 0.75 \mathrm {~m}$, with $A$ between $O$ and $B$. A particle $P$ of mass $m$ oscillates on $l$ in simple harmonic motion with centre $O$. The ratio of the kinetic energy of $P$ when it is at $A$ to its kinetic energy when it is at $B$ is $12 : 11$. Find the amplitude of the motion.

Given that the greatest speed of $P$ is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the time taken by $P$ to travel directly from $A$ to $B$.

\hfill \mbox{\textit{CAIE FP2 2014 Q2 [8]}}
This paper (12 questions)
View full paper
Q1 2 Q2 8 Q3 10 Q4 10 Q5 13 Q6 5 Q7 8 Q8 9 Q9 10 Q10 11 Q11 EITHER Q11 OR