A2 October 2021 Q7
7. [In this question, you may assume that the centre of mass of a circular arc, radius \(r\), with angle at centre \(2\alpha\), is a distance \(\dfrac{r\sin\alpha}{\alpha}\) from the centre.]
A thin non-uniform metal plate is in the shape of a sector \(OAB\) of a circle with centre \(O\) and radius \(a\). The angle \(AOB = \dfrac{\pi}{2}\), as shown in Figure 5.
The plate is modelled as a non-uniform lamina.
The mass per unit area of the lamina, at any point \(P\) of the lamina, is modelled as \(k(OP)^2\), where \(k = \dfrac{4\lambda}{\pi a^4}\) and \(\lambda\) is a constant.
Using the model,
| Scheme | Marks | AO |
|---|---|---|
| Use of an appropriate element (quarter of a circle) | M1 | 2.1 |
| \(\delta A \simeq \dfrac{1}{2}\pi x\delta x\) | A1 | 1.1b |
| \(\delta m \simeq \dfrac{1}{2}\pi x\delta x\times\dfrac{4\lambda}{\pi a^4}x^2 \quad \left(= \dfrac{2\lambda}{a^4}x^3\delta x\right)\) | A1 | 3.4 |
| \(M = \displaystyle\int_0^a \dfrac{2\lambda}{a^4}x^3\,\mathrm{d}x\) | M1 | 2.1 |
| \(M = \dfrac{1}{2}\lambda\) | A1 | 1.1b |
| (5) |
Notes
M1: Use of an appropriate element (may be implied)
A1: Correct expression for area of element (may be implied)
A1: Use of proportionality model to obtain mass of element (may be implied)
M1: Integrating with correct limits
A1: cao
| Scheme | Marks | AO |
|---|---|---|
| Use of “\(\bar{x} = \dfrac{1}{M}\displaystyle\int x\,\mathrm{d}m\)” | M1 | 3.4 |
| \(= \dfrac{1}{M}\displaystyle\int_0^a \left(\dfrac{2\sqrt{2}x}{\pi}\right)\dfrac{2\lambda}{a^4}x^3\,\mathrm{d}x\) | A1 | 1.1b |
| Substitute for \(M\), integrate and sub. in limits | M1 | 3.4 |
| \(\bar{x} = \dfrac{8\sqrt{2}a}{5\pi}\) | A1 | 1.1b |
| (4) | ||
| (9 marks) |
Notes
M1: Use the model and correct method
A1: Correct integral
M1: Use the model to complete the equation
A1: cao