Four coplanar forces act at a point. The magnitudes of the forces are FN, 2FN, 3FN and 30N. 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 θ. [Figure 4]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

1. [*M1] For resolving in either direction. Correct number of terms allow sign errors; allow sin/cos mix on θ. Forces that need resolving should be resolved.
2. [A1] F+2Fcos 45=30sin θ
3. [A1] 2Fsin 45+3F=30cosθ
4. [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. If tan(θ) = (2Fsin45+3F)/(F+2Fcos45) so have θ = tan⁻¹((2Fsin45+3F)/(F+2Fcos45)) then allow M1.
5. [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.
6. [A1] F =5.96 [5.96270...] and θ=28.7 [28.6750...]. Awrt to 5.96 and 28.7. Allow 5.97.