CAIE M1 2024 March — Question 4 6 marks

Exam BoardCAIE
ModuleM1 (Mechanics 1)
Year2024
SessionMarch
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicForces, equilibrium and resultants
TypeEquilibrium of particle under coplanar forces
DifficultyModerate -0.3 This is a standard equilibrium problem requiring resolution of forces in two perpendicular directions and solving simultaneous equations. While it involves multiple forces and an unknown angle, the method is routine for M1 students: resolve horizontally and vertically, then solve. The algebra is straightforward, making it slightly easier than average.
Spec3.03m Equilibrium: sum of resolved forces = 03.03n Equilibrium in 2D: particle under forces
\includegraphics{figure_4} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(2F\) N, \(3F\) N and \(30\) N. The directions of the forces are as shown in the diagram. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]
Question 4:
AnswerMarks Guidance
4For resolving in either direction *M1
sin/cos mix on θ.
Forces that need resolving should be resolved.
AnswerMarks
F+2Fcos45=30sinA1
2Fsin45+3F=30cosA1
1+2cos45
=tan−1
 
2sin45+3
F +2Fcos45
or =sin−1  
 30 
2Fsin45+3F 
or =cos−1  
AnswerMarks Guidance
 30 DM1 For attempt to find θ.
Using their F which can be solved for θ.
From equations with correct number of relevant
terms, forces that need resolving should be resolved.
cos 2sin45+3
If tan= , so have =tan−1  ,
sin 1+2cos45
then allow M1.
 F2(3+2sin45)2 +F2(1+2cos45)2 =302 
 
302
F =
(3+2sin45)2 +(1+2cos45)2
30sin
or F =
1+2cos45
30cos
or F =
AnswerMarks Guidance
3+2sin45DM1 For attempt to find F.
From equations with the correct number of relevant
terms, forces that need resolving should be resolved.
Using their θ.
AnswerMarks Guidance
F =5.96 [5.96270…] and =28.7 [28.6750…]A1 Awrt to 5.96 and 28.7.
Allow 5.97.
6
AnswerMarks Guidance
QuestionAnswer Marks 
Question 4:
4 | For resolving in either direction | *M1 | Correct number of terms allow sign errors; allow
sin/cos mix on θ.
Forces that need resolving should be resolved.
F+2Fcos45=30sin | A1
2Fsin45+3F=30cos | A1
1+2cos45
=tan−1
 
2sin45+3
F +2Fcos45
or =sin−1  
 30 
2Fsin45+3F 
or =cos−1  
 30  | DM1 | For attempt to find θ.
Using their F which can be solved for θ.
From equations with correct number of relevant
terms, forces that need resolving should be resolved.
cos 2sin45+3
If tan= , so have =tan−1  ,
sin 1+2cos45
then allow M1.
 F2(3+2sin45)2 +F2(1+2cos45)2 =302 
 
302
F =
(3+2sin45)2 +(1+2cos45)2
30sin
or F =
1+2cos45
30cos
or F =
3+2sin45 | DM1 | For attempt to find F.
From equations with the correct number of relevant
terms, forces that need resolving should be resolved.
Using their θ.
F =5.96 [5.96270…] and =28.7 [28.6750…] | A1 | Awrt to 5.96 and 28.7.
Allow 5.97.
6
Question | Answer | Marks | Guidance
\includegraphics{figure_4}

Four coplanar forces act at a point. The magnitudes of the forces are $F$ N, $2F$ N, $3F$ N and $30$ N. The directions of the forces are as shown in the diagram.

Given that the forces are in equilibrium, find the value of $F$ and the value of $\theta$. [6]

\hfill \mbox{\textit{CAIE M1 2024 Q4 [6]}}
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