M1 January 2010 Q4
4.
A pole \(AB\) has length 3 m and weight \(W\) newtons. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole at the points \(A\) and \(C\) where \(AC = 1.8\) m, as shown in Figure 2. A load of weight 20 N is attached to the rod at \(B\). The pole is modelled as a uniform rod, the ropes as light inextensible strings and the load as a particle.
(a) Show that the tension in the rope attached to the pole at \(C\) is \(\left(\dfrac{5}{6}W + \dfrac{100}{3}\right)\) N. (4)
(b) Find, in terms of \(W\), the tension in the rope attached to the pole at \(A\). (3)
Given that the tension in the rope attached to the pole at \(C\) is eight times the tension in the rope attached to the pole at \(A\),
(c) find the value of \(W\). (3)
| Scheme | Marks |
|---|---|
| M\((A)\) \(W \times 1.5 + 20 \times 3 = Y \times 1.8\) | M1 A2 (1, 0) |
| \(Y = \dfrac{5}{6}W + \dfrac{100}{3}\) * cso | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\uparrow\) \(X + Y = W + 20\) or equivalent | M1 A1 |
| \(X = \dfrac{1}{6}W - \dfrac{40}{3}\) | A1 |
| (3) |
Alternative to (b)
| M\((C)\) \(X \times 1.8 + 20 \times 1.2 = W \times 0.3\) | M1 A1 |
| \(X = \dfrac{1}{6}W - \dfrac{40}{3}\) | A1 |
| Scheme | Marks |
|---|---|
| \(\dfrac{5}{6}W + \dfrac{100}{3} = 8\left(\dfrac{1}{6}W - \dfrac{40}{3}\right)\) | M1 A1 ft |
| \(W = 280\) | A1 |
| (3) | |
| (10 marks) |