| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Small oscillations period |
| Difficulty | Challenging +1.2 This is a standard compound pendulum problem from Further Maths requiring calculation of moment of inertia using parallel axis theorem and derivation of SHM equation for small oscillations. While it involves multiple steps and Further Maths content (making it harder than average A-level), the techniques are routine applications of standard formulas with 'show that' scaffolding providing the target answers. |
| Spec | 6.04a Centre of mass: gravitational effect6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(I_A = \frac{1}{2} \cdot 3M(2a)^2 \quad [= 6Ma^2]\) | *M1 | Find MI of \(A\) // to axis \(l\) at \(A\)'s centre by \(\perp\) axis theorem |
| \(I_A' = I_A + 3M(2a)^2 \quad [= 18Ma^2]\) (dep *M1) | M1 A1 | Find MI of \(A\) about axis \(l\) |
| \(I_B = \frac{1}{2} \cdot 2M(3a)^2 \quad [= 9Ma^2]\) | **M1 | Find MI of \(B\) (or \(C\)) // to axis \(l\) at its centre by \(\perp\) axis theorem |
| \(I_B' = I_B + 2M(6a)^2 \quad [= 81Ma^2]\) (dep **M1) | M1 A1 | Find MI of \(B\) (or \(C\)) about axis \(l\) |
| \(I = (18 + 2\times 81)Ma^2 = 180Ma^2\) AG | A1 | Verify MI of object about axis \(l\) (A0 if inadequate explanation) |
| SC: \(I_A' = \frac{1}{2}\{3M(2a)^2 + 3M(2a)^2\} \quad [= 12Ma^2]\) | (M1) | SC: Invalidly applying theorems in wrong order |
| \(I_B' = \frac{1}{2}\{2M(3a)^2 + 2M(6a)^2\} \quad [= 45Ma^2]\) | (M1) | \(\left(\max \frac{2}{7}\right)\) |
| 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([-]\, I\dfrac{d^2\theta}{dt^2} = 3Mg \times 2a\sin\theta + 2 \times 2Mg \times 6a\sin\theta\) *or* \(7Mg \times \dfrac{30a}{7}\sin\theta \quad [= 30Mga\sin\theta]\) | *M1 A1 | Use eqn of circular motion to find \(d^2\theta/dt^2\) where \(\theta\) is angle of plane of object with vertical |
| \(\dfrac{d^2\theta}{dt^2} = -\left(\dfrac{g}{6a}\right)\theta\) *or* \(-\left(\dfrac{0.167g}{a}\right)\theta\) (M1 dep *M1) | M1 A1 | Approximate \(\sin\theta\) by \(\theta\) to show SHM (no '\(-\)' is M1 A0) |
| \(T = \dfrac{2\pi}{\sqrt{\left(\dfrac{g}{6a}\right)}} = 2\pi\sqrt{\dfrac{6a}{g}}\) *or* \(15.4\sqrt{\dfrac{a}{g}}\) *or* \(4.87\sqrt{a}\) (AEF) | B1\(\sqrt{}\) | Find period \(T\) from \(T = 2\pi/\omega\) (FT on \(\omega^2\); requires some simplification) |
| 5 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $I_A = \frac{1}{2} \cdot 3M(2a)^2 \quad [= 6Ma^2]$ | *M1 | Find MI of $A$ // to axis $l$ at $A$'s centre by $\perp$ axis theorem |
| $I_A' = I_A + 3M(2a)^2 \quad [= 18Ma^2]$ (dep *M1) | M1 A1 | Find MI of $A$ about axis $l$ |
| $I_B = \frac{1}{2} \cdot 2M(3a)^2 \quad [= 9Ma^2]$ | **M1 | Find MI of $B$ (or $C$) // to axis $l$ at its centre by $\perp$ axis theorem |
| $I_B' = I_B + 2M(6a)^2 \quad [= 81Ma^2]$ (dep **M1) | M1 A1 | Find MI of $B$ (or $C$) about axis $l$ |
| $I = (18 + 2\times 81)Ma^2 = 180Ma^2$ **AG** | A1 | Verify MI of object about axis $l$ (A0 if inadequate explanation) |
| **SC:** $I_A' = \frac{1}{2}\{3M(2a)^2 + 3M(2a)^2\} \quad [= 12Ma^2]$ | (M1) | **SC:** Invalidly applying theorems in wrong order |
| $I_B' = \frac{1}{2}\{2M(3a)^2 + 2M(6a)^2\} \quad [= 45Ma^2]$ | (M1) | $\left(\max \frac{2}{7}\right)$ |
| | **7** | |
---
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[-]\, I\dfrac{d^2\theta}{dt^2} = 3Mg \times 2a\sin\theta + 2 \times 2Mg \times 6a\sin\theta$ *or* $7Mg \times \dfrac{30a}{7}\sin\theta \quad [= 30Mga\sin\theta]$ | *M1 A1 | Use eqn of circular motion to find $d^2\theta/dt^2$ where $\theta$ is angle of plane of object with vertical |
| $\dfrac{d^2\theta}{dt^2} = -\left(\dfrac{g}{6a}\right)\theta$ *or* $-\left(\dfrac{0.167g}{a}\right)\theta$ (M1 dep *M1) | M1 A1 | Approximate $\sin\theta$ by $\theta$ to show SHM (no '$-$' is M1 A0) |
| $T = \dfrac{2\pi}{\sqrt{\left(\dfrac{g}{6a}\right)}} = 2\pi\sqrt{\dfrac{6a}{g}}$ *or* $15.4\sqrt{\dfrac{a}{g}}$ *or* $4.87\sqrt{a}$ (AEF) | B1$\sqrt{}$ | Find period $T$ from $T = 2\pi/\omega$ (FT on $\omega^2$; requires some simplification) |
| | **5** | |(i) Show that the moment of inertia of the object about the axis $l$ is $180 M a ^ { 2 }$.\\
(ii) Show that small oscillations of the object about the axis $l$ are approximately simple harmonic, and state the period.\\
\hfill \mbox{\textit{CAIE FP2 2018 Q5 [12]}}