M5 June 2016 Q4
4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius \(r\), height \(h\) and mass \(M\), about a diameter of one end. (10)
| Scheme | Marks |
|---|---|
| \(\delta A = 2\pi r\delta x\) | M1 |
| \(\delta m = 2\pi r\delta x \cdot \dfrac{M}{2\pi rh}\ \ \left(= \dfrac{M\delta x}{h}\right)\) | M1 A1 (\(\rho\)) |
| \(\dfrac{1}{2}\delta mr^2\) | B1 |
| \(\delta I = \dfrac{1}{2}\delta mr^2 + \delta mx^2\) | M1 A1 |
| \(= \dfrac{M\delta x}{2h}(r^2 + 2x^2)\) | A1 |
| \(I = \displaystyle\int_0^h \frac{M}{2h}(r^2 + 2x^2)\,\mathrm{d}x\) | DM1 |
| \(= \dfrac{M}{2h}\left[\left(r^2x + \tfrac{2}{3}x^3\right)\right]_0^h\) | A1 |
| \(= \dfrac{M}{6}(3r^2 + 2h^2)\) | A1 |
| (10 marks) |
Notes
First M1 for area of hoop (element)
Second M1 for finding the mass of their element by multiplying by the mass per unit area (or by a calculated \(\rho\))
First A1 for a correct mass per unit area (appropriate \(\rho\))
First B1 for correct MI about diameter of hoop
Third M1 for use of parallel axes
Second A1 for correct in terms of \(\delta m\)
Third A1 for correct MI in terms of \(x\)
Fourth DM1 dependent on third M1 for integrating
Fourth A1 for a correct expression with correct limits
Fifth A1 for a correct answer in any form
N.B. The first 8 marks are available for misreads of solid cylinder or cylindrical shell with end(s).