M3 June 2010 Q1
1.
A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length \(13l\). The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find
(a) the tension in the string, (3)
(b) the speed of the ball. (4)
| Scheme | Marks |
|---|---|
| \(\cos\alpha = \dfrac{12}{13}\) | B1 |
| R\((\uparrow)\quad T\cos\alpha = mg\) | M1 |
| \(T \times \dfrac{12}{13} = mg\) | |
| \(T = \dfrac{13}{12}mg\) oe | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Eqn of motion \(\quad T\sin\alpha = m\dfrac{v^2}{5l}\) | M1 A1 |
| \(\dfrac{13mg}{12} \times \dfrac{5}{13} = m\dfrac{v^2}{5l}\) | M1 dep |
| \(v^2 = \dfrac{25gl}{12}\) | |
| \(v = \dfrac{5}{2}\sqrt{\dfrac{gl}{3}} \qquad \left(\text{accept } 5\sqrt{\dfrac{gl}{12}} \text{ or } \sqrt{\dfrac{25gl}{12}} \text{ or any other equiv}\right)\) | A1 |
| (4) | |
| (7 marks) |