M5 June 2018 Q4
4. A uniform lamina of mass \(M\) kg is modelled as the region which is bounded by the curve with equation \(y = x^2\), the positive \(x\)-axis and the line with equation \(x = 2\). The unit of length on both axes is the metre. Find the moment of inertia of the lamina about the \(x\)-axis. (6)
| Scheme | Marks |
|---|---|
| \(\displaystyle\int_0^2 x^2\,\mathrm{d}x = \dfrac{8}{3}\) | M1 |
| \(\rho = \dfrac{3M}{8}\) | A1 |
| \(\delta m = y\rho\,\delta x\) | |
| \(= \dfrac{3Mx^2\,\delta x}{8}\) | M1 |
| \(\delta I = \dfrac{1}{3}\delta m y^2\) | |
| \(= \dfrac{M}{8}x^6\,\delta x\) | M1 |
| \(I = \dfrac{M}{8}\displaystyle\int_0^2 x^6\,\mathrm{d}x = \dfrac{16M}{7}\) kg m\(^2\) | M1 A1 |
| (6) |
Notes
First M1 for finding the area of the lamina
First A1 for a correct mass per unit area (\(\rho\))
Second M1 for an expression for the mass of the strip in terms of \(M\) and \(x\) only.
Third M1 for a CORRECT expression for the MI of the strip in terms of \(M\) and \(x\) only.
Fourth M1 for summing these MI’s from \(x = 0\) to \(x = 2\)
Second A1 for the answer