CAIE M1 2016 June — Question 3 6 marks

Exam BoardCAIE
ModuleM1 (Mechanics 1)
Year2016
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMoments
TypeCoplanar forces in equilibrium
DifficultyModerate -0.3 This is a standard two-unknown equilibrium problem requiring resolution of forces in two perpendicular directions and solving simultaneous equations. While it involves some trigonometry and algebraic manipulation, it follows a routine procedure taught in M1 with no novel problem-solving required, making it slightly easier than average.
Spec3.03m Equilibrium: sum of resolved forces = 03.03n Equilibrium in 2D: particle under forces
3 \includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767} The coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).
Question 3:
AnswerMarks Guidance
AnswerMark Guidance
M1For resolving forces horizontally and/or vertically
\(12\cos75° + P\cos\theta° = 18\cos65°\)A1
\(18\sin65° + 12\sin75° = 15 + P\sin\theta°\)A1
\([P^2 = (18\sin65° + 12\sin75° - 15)^2 + (18\cos65° - 12\cos75°)^2]\) or \([\theta = \tan^{-1}(18\sin65° + 12\sin75° - 15)/(18\cos65° - 12\cos75°)]\)M1 For eliminating either \(\theta\) or \(P\) from the simultaneous equations
\(P = 13.7\) or \(\theta = 70.8\)A1
\(\theta = 70.8\) or \(P = 13.7\)B1 [6 marks] 
## Question 3:

| Answer | Mark | Guidance |
|--------|------|----------|
| | M1 | For resolving forces horizontally and/or vertically |
| $12\cos75° + P\cos\theta° = 18\cos65°$ | A1 | |
| $18\sin65° + 12\sin75° = 15 + P\sin\theta°$ | A1 | |
| $[P^2 = (18\sin65° + 12\sin75° - 15)^2 + (18\cos65° - 12\cos75°)^2]$ or $[\theta = \tan^{-1}(18\sin65° + 12\sin75° - 15)/(18\cos65° - 12\cos75°)]$ | M1 | For eliminating either $\theta$ or $P$ from the simultaneous equations |
| $P = 13.7$ or $\theta = 70.8$ | A1 | |
| $\theta = 70.8$ or $P = 13.7$ | B1 | **[6 marks]** |

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3\\
\includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767}

The coplanar forces shown in the diagram are in equilibrium. Find the values of $P$ and $\theta$.

\hfill \mbox{\textit{CAIE M1 2016 Q3 [6]}}
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