M5 June 2013 Q6
6.
A light inextensible string has a particle of mass \(m\) attached to one end and a particle of mass \(4m\) attached to the other end. The string passes over a rough pulley which is modelled as a uniform circular disc of radius \(a\) and mass \(2m\), as shown in Figure 2.
The pulley can rotate in a vertical plane about a fixed horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. As the pulley rotates, a frictional couple of constant magnitude \(2mga\) acts on it.
The system is held with the string vertical and taut on each side of the pulley and released from rest. Given that the string does not slip on the pulley, find the initial angular acceleration of the pulley. (10)
| Scheme | Marks |
|---|---|
| \(4mg - T_1 = 4mf\) | M1 A1 |
| \(T_2 - mg = mf\) | M1 A1 |
| \((T_1 - T_2)a - 2mga = \dfrac{1}{2}2ma^2.\dfrac{f}{a}\) | M1 A3 |
| \(4mg - mg - 2mg = 6mf \Rightarrow f = \dfrac{g}{6}\) | DM1 |
| ang accln \(= \dfrac{g}{6a}\) | A1 |
| (10 marks) |
Notes
First M1 for equation of motion for \(4m\)
First A1 for correct equation using either \(f\) or \(\alpha\)
Second M1 for equation of motion for \(m\)
Second A1 for correct equation using either \(f\) or \(\alpha\)
Third M1 for equation of motion for pulley
A3 for a correct equation using either \(f\) or \(\alpha\)
Fourth M1, dependent on previous three M’s, for producing an equation in \(\alpha\) and g only.
Sixth A1 for answer
S.C. M4A5 ‘whole system’ equation: \((4mg - mg - 2mg)a = (4ma^2 + ma^2 + ma^2)\alpha\)