| Exam Board | OCR |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2003 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Solid of revolution MI |
| Difficulty | Challenging +1.2 This is a standard M4/Further Mechanics question requiring three routine integrations for a solid of revolution. Part (i) is a 'show that' for mass using volume integration, part (ii) requires finding the centre of mass using a standard formula, and part (iii) applies the moment of inertia formula. While it involves multiple steps and Further Maths content (making it harder than typical A-level), the techniques are direct applications of standard formulas with straightforward integration of power functions. No novel insight or complex manipulation is required. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks | Guidance |
|---|---|---|
| mass \(= 350\int_0^8 \pi x^4 dx = 350\pi \left[\frac{1}{5}x^5\right]_0^8 = 210\pi \times 32 = 6720\pi\) | M1 A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = \frac{3360}{6720} = 5\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(I = \int_0^8 175\pi x^8 dx = 175\pi\left[\frac{1}{9}x^9\right]_0^8 = 75\pi \times 128 = 9600\pi\) kg m² | M1 A1 | [4] |
**Part (a): Mass**
mass $= 350\int_0^8 \pi x^4 dx = 350\pi \left[\frac{1}{5}x^5\right]_0^8 = 210\pi \times 32 = 6720\pi$ | M1 A1 | [3]
**Part (b): Centre of Mass**
mass of 'elemental disc' $= \rho\pi y^2 \delta x = 350\pi x^4 \delta x$
$6720\pi\bar{x} = \int_0^8 x(350\pi x^4) dx = 350\pi\left[\frac{1}{6}x^6\right]_0^8 = \frac{35}{6}\pi \times 256$
$\bar{x} = \frac{3360}{6720} = 5$ | A1 | [3]
**Part (c): Second Moment of Inertia**
M.O.I. of 'elemental disc' $= \frac{1}{2}mr^2 = \frac{1}{2}(350\pi x^4 \delta x)x^4 = 175\pi x^8 \delta x$
$I = \int_0^8 175\pi x^8 dx = 175\pi\left[\frac{1}{9}x^9\right]_0^8 = 75\pi \times 128 = 9600\pi$ kg m² | M1 A1 | [4]5 The region bounded by the $x$-axis, the line $x = 8$ and the curve $y = x ^ { \frac { 1 } { 3 } }$ for $0 \leqslant x \leqslant 8$, is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid of revolution. The unit of length is the metre, and the density of the solid is $350 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$.\\
(i) Show that the mass of the solid is $6720 \pi \mathrm {~kg}$.\\
(ii) Find the $x$-coordinate of the centre of mass of the solid.\\
(iii) Find the moment of inertia of the solid about the $x$-axis.
\hfill \mbox{\textit{OCR M4 2003 Q5 [10]}}