M2 June 2009 Q4
4.
A uniform rod \(AB\), of length 1.5 m and mass 3 kg, is smoothly hinged to a vertical wall at \(A\). The rod is held in equilibrium in a horizontal position by a light strut \(CD\) as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The end \(C\) of the strut is freely jointed to the wall at a point 0.5 m vertically below \(A\). The end \(D\) is freely joined to the rod so that \(AD\) is 0.5 m.
(a) Find the thrust in \(CD\). (4)
(b) Find the magnitude and direction of the force exerted on the rod \(AB\) at \(A\). (7)
| Scheme | Marks |
|---|---|
| Taking moments about A: \(3g \times 0.75 = \dfrac{T}{\sqrt{2}} \times 0.5\) | M1A1A1 |
| \(T = 3\sqrt{2}g \times \dfrac{7.5}{5} = \dfrac{9\sqrt{2}g}{2}\ (= 62.4N)\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\leftarrow\ \pm H = \dfrac{T}{\sqrt{2}}\ \left(= \dfrac{9g}{2} \approx 44.1N\right)\) | B1 |
| \(\uparrow\ \pm V + \dfrac{T}{\sqrt{2}} = 3g\ \ \left(\Rightarrow V = 3g - \dfrac{9g}{2} = \dfrac{-3g}{2} \approx -14.7\text{ N}\right)\) | M1A1 |
| \(\Rightarrow |R| = \sqrt{81 + 9} \times \dfrac{g}{2} \approx 46.5(N)\) | M1A1 |
| at angle \(\tan^{-1}\dfrac{1}{3} = 18.4^\circ\) (0.322 radians) below the line of BA \(161.6^\circ\) (2.82 radians) below the line of AB (\(108.4^\circ\) or 1.89 radians to upward vertical) | M1A1 |
| (7) | |
| (11 marks) |