CAIE FP2 2008 November — Question 3 9 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2008
SessionNovember
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMoments
TypeRotation about fixed axis: angular acceleration and velocity
DifficultyChallenging +1.2 This is a standard rotation problem requiring application of Newton's second law for both linear and rotational motion, combined with the parallel axis theorem or moment of inertia formula for a disc. While it involves multiple connected concepts (torque, tension, resistance forces, energy or equations of motion), the setup is straightforward and the solution follows a well-practiced method. The multi-step nature and need to handle both parts (angular acceleration and angular speed after one revolution) elevates it slightly above average difficulty, but it remains a textbook-style mechanics question without requiring novel insight.
Spec6.01c Dimensional analysis: error checking6.04d Integration: for centre of mass of laminas/solids
3 \includegraphics[max width=\textwidth, alt={}, center]{28e7fb78-e2b6-4f6e-92dc-a06eb87fe1ef-2_582_513_1292_815} A uniform disc, of mass \(m\) and radius \(a\), is free to rotate without resistance in a vertical plane about a horizontal axis through its centre. A light inextensible string has one end fixed to the rim of the disc, and is wrapped round the rim. A block of mass \(2 m\) is attached to the other end of the string (see diagram). The system is released from rest with the block hanging vertically. While the block moves it experiences a constant resistance to motion of magnitude \(\frac { 1 } { 10 } m g\). Find the angular acceleration of the disc, and find also the angular speed of the disc when it has turned through one complete revolution.
[0pt] [9]
Question 3:
AnswerMarks Guidance
Answer/WorkingMark Guidance
EITHER: \(2ma\,d^2\theta/dt^2 = 2mg - T - mg/10\)M1 A1 Relate angular accel. to tension for block
\(I\,d^2\theta/dt^2 = aT\), \(I = \frac{1}{2}ma^2\)M1 A1 Relate angular accel. to tension for disc
\((\frac{1}{2} + 2)ma^2\,d^2\theta/dt^2 = (2 - 0.1)mga\)M1 Eliminate tension \(T\)
\(d^2\theta/dt^2 = 19g/25a\) or \(0.76g/a\) or \(7.6/a\)A1
\((d\theta/dt)^2 = 2\,d^2\theta/dt^2 \cdot 2\pi\) (\(\sqrt{}\) on \(d^2\theta/dt^2\)): \((d\theta/dt)^2 = 76\pi g/25a\) A.E.F.M1 A1\(\sqrt{}\)
\(d\theta/dt = 3.09\sqrt{(g/a)}\) or \(9.77/\sqrt{a}\)A1 Find \(d\theta/dt\) (A.E.F.)
OR: \(\frac{1}{2}I(d\theta/dt)^2 + \frac{1}{2}\cdot 2m(a\,d\theta/dt)^2\)(M1 A1) Use conservation of energy for system
\(= 2mga\theta - 0.1\,mga\theta\)(M1 A1)
Put \(\theta = 2\pi\) and find \(d\theta/dt\) (A.E.F.): \(d\theta/dt = 3.09\sqrt{(g/a)}\) or \(9.77/\sqrt{a}\)(M1 A1)
Differentiate energy eqn w.r.t. \(t\): \((5ma^2/4)\cdot 2\,d^2\theta/dt^2 = 1.9\,mga\)(M1 A1)
\(d^2\theta/dt^2 = 19g/25a\) or \(0.76g/a\) or \(7.6/a\)(A1) Total: 9 
## Question 3:

| Answer/Working | Mark | Guidance |
|---|---|---|
| EITHER: $2ma\,d^2\theta/dt^2 = 2mg - T - mg/10$ | M1 A1 | Relate angular accel. to tension for block |
| $I\,d^2\theta/dt^2 = aT$, $I = \frac{1}{2}ma^2$ | M1 A1 | Relate angular accel. to tension for disc |
| $(\frac{1}{2} + 2)ma^2\,d^2\theta/dt^2 = (2 - 0.1)mga$ | M1 | Eliminate tension $T$ |
| $d^2\theta/dt^2 = 19g/25a$ or $0.76g/a$ or $7.6/a$ | A1 | |
| $(d\theta/dt)^2 = 2\,d^2\theta/dt^2 \cdot 2\pi$ ($\sqrt{}$ on $d^2\theta/dt^2$): $(d\theta/dt)^2 = 76\pi g/25a$ A.E.F. | M1 A1$\sqrt{}$ | |
| $d\theta/dt = 3.09\sqrt{(g/a)}$ or $9.77/\sqrt{a}$ | A1 | Find $d\theta/dt$ (A.E.F.) |
| OR: $\frac{1}{2}I(d\theta/dt)^2 + \frac{1}{2}\cdot 2m(a\,d\theta/dt)^2$ | (M1 A1) | Use conservation of energy for system |
| $= 2mga\theta - 0.1\,mga\theta$ | (M1 A1) | |
| Put $\theta = 2\pi$ and find $d\theta/dt$ (A.E.F.): $d\theta/dt = 3.09\sqrt{(g/a)}$ or $9.77/\sqrt{a}$ | (M1 A1) | |
| Differentiate energy eqn w.r.t. $t$: $(5ma^2/4)\cdot 2\,d^2\theta/dt^2 = 1.9\,mga$ | (M1 A1) | |
| $d^2\theta/dt^2 = 19g/25a$ or $0.76g/a$ or $7.6/a$ | (A1) | **Total: 9** |

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\includegraphics[max width=\textwidth, alt={}, center]{28e7fb78-e2b6-4f6e-92dc-a06eb87fe1ef-2_582_513_1292_815}

A uniform disc, of mass $m$ and radius $a$, is free to rotate without resistance in a vertical plane about a horizontal axis through its centre. A light inextensible string has one end fixed to the rim of the disc, and is wrapped round the rim. A block of mass $2 m$ is attached to the other end of the string (see diagram). The system is released from rest with the block hanging vertically. While the block moves it experiences a constant resistance to motion of magnitude $\frac { 1 } { 10 } m g$. Find the angular acceleration of the disc, and find also the angular speed of the disc when it has turned through one complete revolution.\\[0pt]
[9]

\hfill \mbox{\textit{CAIE FP2 2008 Q3 [9]}}
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Q1 5 Q2 8 Q3 9 Q4 10 Q5 11 Q6 3 Q7 8 Q8 9 Q9 10 Q10 13 Q11 EITHER Q11 OR