Question 2 - A-Level Maths

CAIE M1 2017 June — Question 2 5 marks

Exam Board	CAIE
Module	M1 (Mechanics 1)
Year	2017
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Moments
Type	Coplanar forces in equilibrium
Difficulty	Moderate -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's a routine mechanics exercise with a clear method that M1 students practice extensively. Slightly easier than average due to being a straightforward application of equilibrium conditions without requiring moments or geometric insight.
Spec	3.03m Equilibrium: sum of resolved forces = 0

2 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-03_522_604_262_769} The four coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).

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Question 2:

Answer	Marks	Guidance
Answer	Marks	Guidance
EITHER:
\(3P\sin55 + P\sin\theta = 20 + P\sin\theta\) or \(3P\sin55 = 20\)	(M1)	Resolves forces vertically
\(P = 8.14\)	A1
\(3P\cos55 = 2P\cos\theta\)	M1	Resolves forces horizontally
\(\cos\theta = 1.5\cos55 \rightarrow \theta = \ldots\)	M1	Attempt to solve for \(\theta\)
\(\theta = 30.6\)	A1)
OR:
\(\dfrac{3P}{\sin90} = \dfrac{20}{\sin125}\)	(M1)	Uses Lami's Theorem (forces \(3P\) and \(20\))
\(P = 8.14\)	A1
\(\dfrac{3P}{\sin90} = \dfrac{2P\cos\theta}{\sin145}\)	M1	Uses Lami's Theorem (forces \(3P\) and \(2P\cos\theta\))
\(\cos\theta = 1.5\sin145 \rightarrow \theta = \ldots\)	M1	Attempt to solve for \(\theta\)
\(\theta = 30.6\)	A1)
Total: 5

## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| *EITHER:* | | |
| $3P\sin55 + P\sin\theta = 20 + P\sin\theta$ or $3P\sin55 = 20$ | (M1) | Resolves forces vertically |
| $P = 8.14$ | A1 | |
| $3P\cos55 = 2P\cos\theta$ | M1 | Resolves forces horizontally |
| $\cos\theta = 1.5\cos55 \rightarrow \theta = \ldots$ | M1 | Attempt to solve for $\theta$ |
| $\theta = 30.6$ | A1) | |
| *OR:* | | |
| $\dfrac{3P}{\sin90} = \dfrac{20}{\sin125}$ | (M1) | Uses Lami's Theorem (forces $3P$ and $20$) |
| $P = 8.14$ | A1 | |
| $\dfrac{3P}{\sin90} = \dfrac{2P\cos\theta}{\sin145}$ | M1 | Uses Lami's Theorem (forces $3P$ and $2P\cos\theta$) |
| $\cos\theta = 1.5\sin145 \rightarrow \theta = \ldots$ | M1 | Attempt to solve for $\theta$ |
| $\theta = 30.6$ | A1) | |
| **Total: 5** | | |

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2\\
\includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-03_522_604_262_769}

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

\hfill \mbox{\textit{CAIE M1 2017 Q2 [5]}}

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