M5 June 2005 Q7
7. A uniform lamina of mass \(m\) is in the shape of an equilateral triangle \(ABC\) of perpendicular height \(h\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) through \(A\) and perpendicular to the lamina.
(a) Show, by integration, that the moment of inertia of the lamina about \(L\) is \(\dfrac{5}{9}mh^2\). (9)
The centre of mass of the lamina is \(G\). The lamina is in equilibrium, with \(G\) below \(A\), when it is given an angular speed \(\sqrt{\left(\dfrac{6g}{5h}\right)}\).
(b) Find the angle between \(AG\) and the downward vertical, when the lamina first comes to rest. (5)
(c) Find the greatest magnitude of the angular acceleration during the motion. (3)
| Scheme | Marks |
|---|---|
| \(\dfrac{l}{x} = \dfrac{b}{h}\ \left(= \dfrac{2}{\sqrt{3}}\right)\) | M1 |
| \(\Rightarrow\ l = \dfrac{bx}{h} = \dfrac{2x}{\sqrt{3}}\) | A1 |
| \(\rho = \dfrac{m}{\frac{1}{2}bh} = \dfrac{2m}{bh}\) | |
| \(\delta m = \dfrac{bx}{h}\cdot\delta x\cdot\dfrac{2m}{bh} = \dfrac{2mx\,\delta x}{h^2}\) | M1 |
| \(\delta I_L \simeq \dfrac{1}{3}\delta m\left(\dfrac{l}{2}\right)^2 + \delta m\,x^2\) | M1 M1 A1 |
| \(= \dfrac{1}{12}(l^2 + 12x^2)\,\delta m\) | |
| \(= \dfrac{1}{12}\left(\dfrac{4x^2}{3} + 12x^2\right)\dfrac{2mx\,\delta x}{h^2}\) | |
| \(= \dfrac{80}{36}\dfrac{m}{h^2}x^3\,\delta x\) | |
| \(= \dfrac{20}{9}\dfrac{m}{h^2}x^3\,\delta x\) | A1 |
| \(I_L = \dfrac{20}{9}\dfrac{m}{h^2}\displaystyle\int_0^h x^3\,\mathrm{d}x\) | M1 |
| \(= \dfrac{20}{9}\dfrac{m}{h^2}\cdot\dfrac{h^4}{4}\) | |
| \(= \dfrac{5mh^2}{9}\) * | A1 |
| (9) |
| Scheme | Marks |
|---|---|
| Energy: \(\dfrac{1}{2}\cdot\dfrac{5mh^2}{9}\dot{\theta}^2 = mg\dfrac{2h}{3}(1 - \cos\theta)\) | |
| \(\Rightarrow\ \dfrac{1}{2}\cdot\dfrac{5mh^2}{9}\cdot\dfrac{6g}{5h} = mg\dfrac{2h}{3}(1 - \cos\theta)\) | M1 A1 A1 |
| \(\cos\theta = \dfrac{1}{2}\) | M1 |
| \(\theta = \dfrac{\pi}{3}\ (60^\circ)\) | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| M(\(A\)): \(-mg\cdot\dfrac{2h}{3}\sin 60 = \dfrac{5mh^2}{9}\ddot{\theta}\) (max \(L^r\) acc\(^n\) when at rest) | M1 A1ft on \(\theta\) |
| \(|\ddot{\theta}|_{\max} = \dfrac{3\sqrt{3}g}{5h}\) | A1 |
| (3) | |
| (17 marks) |