M3 June 2012 Q3
3.

A particle \(Q\) of mass 5 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a vertical pole. Each string has length 0.6 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 1.
Both strings are taut and \(Q\) is moving in a horizontal circle with constant angular speed 10 rad s\(^{-1}\).
Find the tension in
(i) \(AQ\),
(ii) \(BQ\). (10)

| Scheme | Marks |
|---|---|
| \(\cos\theta = \dfrac{0.2}{0.6}\left(= \dfrac{1}{3}\right)\) | B1 |
| Resolve vertically: | M1 |
| \(T_A\cos\theta = T_B\cos\theta + mg\ \ \ \ \left(T_A = T_B + 3mg\right)\) | A2,1,0 |
| Acceleration towards the centre: | M1 |
| \(T_A\sin\theta + T_B\sin\theta = m \times 0.6\sin\theta \times \omega^2\ \ \ \left(T_A + T_B = 5 \times \dfrac{3}{5} \times 100 = 300\right)\) | A2,1,0 |
| Substitute values for \(\omega\) and trig functions and solve to find \(T_A\) or \(T_B\) | M1 |
| \(T_B + 147 + T_B = 300,\ \ \ 2T_B = 300 - 147 = 153\) | A1,A1 |
| \(T_A = 223.5\) (N) , \(T_B = 76.5\) (N) | |
| \(T_A = 224\) or 220 \(T_B = 76\) | |
| \(T_B = 76.5\) or 77 \(T_A = 223\) | |
| (10) | |
| (10 marks) |