Challenging +1.2 This is a multi-step moments problem requiring identification of forces, calculation of perpendicular distances, and solving a trigonometric inequality. While it involves several components (weight, applied force, reaction, and the geometry of a sector), the approach is systematic and follows standard M2 techniques. The sector geometry adds mild complexity, but the question guides students through the setup clearly. It's moderately harder than average due to the combination of moments, geometry, and inequality solving, but remains within standard M2 scope.
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\includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_519_860_1430_641}
\(O A B C\) is the cross-section through the centre of mass of a uniform prism of weight 20 N . The crosssection is in the shape of a sector of a circle with centre \(O\), radius \(O A = r \mathrm {~m}\) and angle \(A O C = \frac { 2 } { 3 } \pi\) radians. The prism lies on a plane inclined at an angle \(\theta\) radians to the horizontal, where \(\theta < \frac { 1 } { 3 } \pi\). OC lies along a line of greatest slope with \(O\) higher than \(C\). The prism is freely hinged to the plane at \(O\). A force of magnitude 15 N acts at \(A\), in a direction towards to the plane and at right angles to it (see diagram). Given that the prism remains in equilibrium, find the set of possible values of \(\theta\).
Accept \(\prec,=,\succ\) as alternative to \(\leq\)
\(\cos(\pi/3 - \theta) \geq 0.68(017...)\)
A1
Accept \(\succ,=,\prec\) as alternative to \(\geq\)
\(\pi/3 - \theta \leq 0.82(279...)\)
M1
Solves for \(\theta\), equation or inequality
\(\theta = 0.224\)
A1
Correct value
\(\theta \geq 0.224\)
A1 [9]
Correct sign, accept \(\succ\); SR deduct 1 mark for assuming \(r = 1\)
Total: [9]
## Question 7:
| Answer | Mark | Guidance |
|--------|------|----------|
| $OG = 2r\sin(\pi/3)/(3\pi/3)$ | B1 | Centre of mass from O |
| $15r\cos(\pi - 2\pi/3)$ | B1 | Moment of 15 N about O |
| $20 \times OG\cos(\pi/3 - \theta)$ | B1ft | Moment of weight about O, ft cv(OG) if used |
| | M1 | Uses moments, including 15 N and 20 N |
| $15r\cos(\pi/3) \leq 20 \times 2r\sin(\pi/3)/\pi \times \cos(\pi/3 - \theta)$ | A1ft | Accept $\prec,=,\succ$ as alternative to $\leq$ |
| $\cos(\pi/3 - \theta) \geq 0.68(017...)$ | A1 | Accept $\succ,=,\prec$ as alternative to $\geq$ |
| $\pi/3 - \theta \leq 0.82(279...)$ | M1 | Solves for $\theta$, equation or inequality |
| $\theta = 0.224$ | A1 | Correct value |
| $\theta \geq 0.224$ | A1 [9] | Correct sign, accept $\succ$; **SR deduct 1 mark for assuming $r = 1$** |
**Total: [9]**
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\includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_519_860_1430_641}\\
$O A B C$ is the cross-section through the centre of mass of a uniform prism of weight 20 N . The crosssection is in the shape of a sector of a circle with centre $O$, radius $O A = r \mathrm {~m}$ and angle $A O C = \frac { 2 } { 3 } \pi$ radians. The prism lies on a plane inclined at an angle $\theta$ radians to the horizontal, where $\theta < \frac { 1 } { 3 } \pi$. OC lies along a line of greatest slope with $O$ higher than $C$. The prism is freely hinged to the plane at $O$. A force of magnitude 15 N acts at $A$, in a direction towards to the plane and at right angles to it (see diagram). Given that the prism remains in equilibrium, find the set of possible values of $\theta$.
\hfill \mbox{\textit{CAIE M2 2013 Q7 [9]}}