| Exam Board | OCR MEI |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Variable density MI integration |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring integration to find moment of inertia with variable density, then applying rotational dynamics with impulse-momentum and energy conservation. The multi-step nature, non-uniform mass distribution, and requirement to find a range of values for complete revolutions elevates this significantly above standard mechanics questions, though the techniques themselves are within the M4 syllabus. |
| Spec | 6.03e Impulse: by a force6.03f Impulse-momentum: relation6.04a Centre of mass: gravitational effect |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(M = k\int_0^{2a}\left(1+\frac{x}{2a}\right)dx = k\left[x + \frac{x^2}{4a}\right]_0^{2a}\) | M1 | Attempt to integrate (limits not required) |
| \(M = 3ka\) | A1 | |
| \(I = k\int_0^{2a}\left(1+\frac{x}{2a}\right)x^2 dx\) | B1 | Limits not required |
| \(= k\left[\frac{x^3}{3} + \frac{x^4}{8a}\right]_0^{2a}\) | M1 | Attempt to integrate and substitute limits |
| \(I = \frac{14}{9}Ma^2\) | E1 | |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{3}{2}aJ = \left(\frac{14}{9}Ma^2\right)\omega\) | M1 | Conservation of angular momentum |
| \(\omega = \frac{27J}{28Ma}\) | A1 | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3ka\bar{x} = k\int_0^{2a}\left(x + \frac{x^2}{2a}\right)dx\) | B1 | Limits not required |
| \(= \left[\frac{x^2}{2} + \frac{x^3}{6a}\right]_0^{2a}\) | M1 | Attempt to integrate (limits not required) to find c of m |
| \(\bar{x} = \frac{10}{9}a\) | A1 | |
| \(-\frac{20}{9}Mga + \frac{1}{2}\left(\frac{14}{9}Ma^2\right)\left(\frac{27J}{28Ma}\right)^2 > 0\) | M1 | Conservation of energy using their \(\bar{x}\) (accept \(\geq\) or \(=\)); must substitute for their \(\omega\) - condone errors in PE term |
| \(J > \frac{8}{27}M\sqrt{35ga}\) | A1 | AEF – accept \(\geq\) |
| [5] |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $M = k\int_0^{2a}\left(1+\frac{x}{2a}\right)dx = k\left[x + \frac{x^2}{4a}\right]_0^{2a}$ | M1 | Attempt to integrate (limits not required) |
| $M = 3ka$ | A1 | |
| $I = k\int_0^{2a}\left(1+\frac{x}{2a}\right)x^2 dx$ | B1 | Limits not required |
| $= k\left[\frac{x^3}{3} + \frac{x^4}{8a}\right]_0^{2a}$ | M1 | Attempt to integrate and substitute limits |
| $I = \frac{14}{9}Ma^2$ | E1 | |
| **[5]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{3}{2}aJ = \left(\frac{14}{9}Ma^2\right)\omega$ | M1 | Conservation of angular momentum |
| $\omega = \frac{27J}{28Ma}$ | A1 | |
| **[2]** | | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3ka\bar{x} = k\int_0^{2a}\left(x + \frac{x^2}{2a}\right)dx$ | B1 | Limits not required |
| $= \left[\frac{x^2}{2} + \frac{x^3}{6a}\right]_0^{2a}$ | M1 | Attempt to integrate (limits not required) to find c of m |
| $\bar{x} = \frac{10}{9}a$ | A1 | |
| $-\frac{20}{9}Mga + \frac{1}{2}\left(\frac{14}{9}Ma^2\right)\left(\frac{27J}{28Ma}\right)^2 > 0$ | M1 | Conservation of energy using their $\bar{x}$ (accept $\geq$ or $=$); must substitute for their $\omega$ - condone errors in PE term |
| $J > \frac{8}{27}M\sqrt{35ga}$ | A1 | AEF – accept $\geq$ |
| **[5]** | | |
---2 A thin rigid rod PQ has length $2 a$. Its mass per unit length, $\rho$, is given by $\rho = k \left( 1 + \frac { x } { 2 a } \right)$ where $x$ is the distance from P and $k$ is a positive constant. The mass of the rod is $M$ and the moment of inertia of the rod about an axis through P perpendicular to PQ is $I$.\\
(i) Show that $I = \frac { 14 } { 9 } M a ^ { 2 }$.
The rod is initially at rest with Q vertically below P . It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through P . The rod is struck a horizontal blow perpendicular to the fixed axis at the point where $x = \frac { 3 } { 2 } a$. The magnitude of the impulse of this blow is $J$.\\
(ii) Find, in terms of $a , J$ and $M$, the initial angular speed of the rod.\\
(iii) Find, in terms of $a , g$ and $M$, the set of values of $J$ for which the rod makes complete revolutions.
\hfill \mbox{\textit{OCR MEI M4 2016 Q2 [12]}}