M5 June 2015 Q5
5. A uniform circular disc, of mass \(m\) and radius \(a\), is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) is a tangent to the disc at the point \(A\). The centre \(O\) of the disc moves in a vertical plane that is perpendicular to \(L\).
The disc is held at rest with its plane horizontal and released.
(a) Find the angular acceleration of the disc when it has turned through an angle of \(\dfrac{\pi}{3}\) (5)
(b) Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis \(L\) acting on the disc at \(A\), when the disc has turned through an angle of \(\dfrac{\pi}{3}\) (4)
| Scheme | Marks |
|---|---|
| \(I_T = \tfrac{1}{4}ma^2 + ma^2\) | M1 |
| \(= \tfrac{5}{4}ma^2\) | A1 |
| \(mga\cos\tfrac{\pi}{3} = \tfrac{5}{4}ma^2\ddot{\theta}\) | M1 A1ft |
| \(\dfrac{2g}{5a} = \ddot{\theta}\) | A1 |
| (5) |
Notes
First M1 for use of perp and parallel axes theorem
First A1 \(5ma^2/4\)
Second M1 for moments about the axis or differentiate a general energy equation
Second A1ft on their \(I\) for correct equation
Third A1 for answer
| Scheme | Marks |
|---|---|
| \(mg\cos\tfrac{\pi}{3} \pm X = ma\ddot{\theta}\) | M1 A1 A1 |
| \(|X| = \tfrac{1}{2}mg - \tfrac{2}{5}mg\) | |
| \(= \tfrac{1}{10}mg\) | A1 |
| (4) | |
| (9 marks) |
Notes
First M1 for resolving along the rod
First A1 A1 for a correct equation (A1 for each side) Need \(\pi/3\)
Third A1 for \(mg/10\) (must be positive)