CAIE FP2 2019 June — Question 5 12 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2019
SessionJune
Marks12
PaperDownload PDF ↗
TopicMoments
TypeRod or object resting on curved surface
DifficultyChallenging +1.8 This is a challenging Further Maths mechanics problem requiring geometric insight about the hemisphere contact, careful resolution in multiple directions, friction at limiting equilibrium, and taking moments about a non-obvious point. The multi-part structure with geometric constraints (hemisphere geometry) and the need to work systematically through forces and moments elevates this significantly above standard A-level mechanics, though the individual techniques are accessible to FM students.
Spec3.04b Equilibrium: zero resultant moment and force6.04e Rigid body equilibrium: coplanar forces
5 \includegraphics[max width=\textwidth, alt={}, center]{34dd6523-7c0c-4842-bbda-56ad8d3f9766-10_456_684_264_731} A uniform \(\operatorname { rod } A B\) of length \(2 x\) and weight \(W\) rests on the smooth rim of a fixed hemispherical bowl of radius \(a\). The end \(B\) of the rod is in contact with the rough inner surface of the bowl. The coefficient of friction between the rod and the bowl at \(B\) is \(\frac { 1 } { 3 }\). A particle of weight \(\frac { 1 } { 4 } W\) is attached to the end \(A\) of the rod. The end \(B\) is about to slip upwards when \(A B\) is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram).
  1. By resolving parallel to the rod, show that the normal component of the reaction of the bowl on the rod at \(B\) is \(\frac { 3 } { 4 } W\).
  2. Find, in terms of \(W\), the reaction between the rod and the smooth rim of the bowl.
  3. Find \(x\) in terms of \(a\).
5\\
\includegraphics[max width=\textwidth, alt={}, center]{34dd6523-7c0c-4842-bbda-56ad8d3f9766-10_456_684_264_731}

A uniform $\operatorname { rod } A B$ of length $2 x$ and weight $W$ rests on the smooth rim of a fixed hemispherical bowl of radius $a$. The end $B$ of the rod is in contact with the rough inner surface of the bowl. The coefficient of friction between the rod and the bowl at $B$ is $\frac { 1 } { 3 }$. A particle of weight $\frac { 1 } { 4 } W$ is attached to the end $A$ of the rod. The end $B$ is about to slip upwards when $A B$ is inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 3 } { 4 }$ (see diagram).\\
(i) By resolving parallel to the rod, show that the normal component of the reaction of the bowl on the rod at $B$ is $\frac { 3 } { 4 } W$.\\

(ii) Find, in terms of $W$, the reaction between the rod and the smooth rim of the bowl.\\

(iii) Find $x$ in terms of $a$.\\

\hfill \mbox{\textit{CAIE FP2 2019 Q5 [12]}}
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Q1 4 Q2 8 Q3 9 Q4 11 Q5 12 Q6 6 Q7 8 Q8 8 Q9 10 Q10 12 Q11 EITHER Q11 OR