M3 January 2011 Q5
5.
A small ball \(P\) of mass \(m\) is attached to the ends of two light inextensible strings of length \(l\). The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(AP\) is perpendicular to \(BP\) as shown in Figure 3. The system rotates about the line \(AB\) with constant angular speed \(\omega\). The ball moves in a horizontal circle.
(a) Find, in terms of \(m\), \(g\), \(l\) and \(\omega\), the tension in \(AP\) and the tension in \(BP\). (8)
(b) Show that \(\omega^2 > \dfrac{g\sqrt{2}}{l}\). (2)
| Scheme | Marks |
|---|---|
| \(r = \dfrac{l}{\sqrt{2}}\) | B1 |
| R\((\uparrow)\quad T_a\cos 45 = T_b\cos 45 + mg\) \(T_a - T_b = mg\sqrt{2} \qquad (1)\) | M1A1 |
| R\((\rightarrow)\quad T_a\cos 45 + T_b\cos 45 = mr\omega^2\) \(T_a \times \dfrac{1}{\sqrt{2}} + T_b \times \dfrac{1}{\sqrt{2}} = m\dfrac{l}{\sqrt{2}}\omega^2\) \(T_a + T_b = ml\omega^2 \qquad (2)\) \(T_a - T_b = mg\sqrt{2} \qquad (1)\) | M1A1 |
| \(2T_a = m\left(l\omega^2 + g\sqrt{2}\right)\) \(T_a = \dfrac{1}{2}m\left(l\omega^2 + g\sqrt{2}\right)\) | M1 |
| \(T_b = ml\omega^2 - T_a\) | A1 |
| \(= \dfrac{1}{2}m\left(l\omega^2 - g\sqrt{2}\right)\) | A1 |
| (8) |
| Scheme | Marks |
|---|---|
| \(T_b > 0 \qquad \dfrac{1}{2}m\left(l\omega^2 - g\sqrt{2}\right) > 0\) | M1 |
| \(\omega^2 > \dfrac{g\sqrt{2}}{l}\) * | A1 |
| (2) | |
| (10 marks) |