M1 June 2014 (R) Q6
6.
A non-uniform beam \(AD\) has weight \(W\) newtons and length 4 m. It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. The ropes are attached to two points \(B\) and \(C\) on the beam, where \(AB = 1\) m and \(CD = 1\) m, as shown in Figure 3. The tension in the rope attached to \(C\) is double the tension in the rope attached to \(B\). The beam is modelled as a rod and the ropes are modelled as light inextensible strings.
A small load of weight \(kW\) newtons is attached to the beam at \(D\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle.
Find
| Scheme | Marks |
|---|---|
| Resolving vertically: \(T + 2T\ (= 3T) = W\) | M1A1 |
| Moments about B: \(2 \times 2T = (d - 1)W\) | M1A1 |
| Substitute and solve for d : \(2 \times 2T = (d - 1)3T\) | DM1 |
| \(d = \dfrac{7}{3}\) (m) | A1 |
| (6) |
Notes
N.B. If \(Wg\) is used, mark as a misread.
First M1 for an equation in \(W\) and \(T\) and possibly \(d\) (either resolve vertically or moments about any point other than the centre of mass of the rod), with usual rules.
First A1 for a correct equation.
Second M1 for an equation in \(W\) and \(T\) and possibly \(d\) (either resolve vertically or moments about any point other than the centre of mass of the rod), with usual rules.
Second A1 for a correct equation.
N.B. The above 4 marks can be scored if their \(d\) is measured from a different point
Third M1, dependent on first and second M marks, for solving for \(d\)
Third A1 for \(d = 7/3\), 2.3 (m) or better
N.B. Alternative
If a single equation is used (see below) by taking moments about the centre of mass of the rod, \(2T(3 - d) = T(d - 1)\), this scores M2A2 (-1 each error)
Third M1, dependent on first and second M marks, for solving for \(d\)
Third A1 for \(d = 7/3\)
| Scheme | Marks |
|---|---|
| Moments about C: \((T_B \times 2) + (kW \times 1) = W \times \dfrac{2}{3}\) | M1A1 |
| \(T_B = W\dfrac{(2 - 3k)}{6}\) or equivalent | A1 |
| (3) |
Notes
First M1 for producing an equation in \(T_B\) and \(W\) only, either by taking moments about \(C\), or using two equations and eliminating
First A1 for a correct equation
Second A1 for \(W(2 - 3k)/6\) oe.
N.B. M0 if they use any information about the tension(s) from part (a).
| Scheme | Marks |
|---|---|
| solving \(T_B \geqslant 0\) or \(T_B > 0\) for \(k\). | M1 |
| \(0 < k \leqslant 2/3\) or \(0 < k < 2/3\) only | A1 |
| (2) | |
| (11 marks) |
Notes
M1 for solving \(T_B \geqslant 0\) or \(T_B > 0\) for \(k\).
A1 for \(0 < k \leqslant 2/3\) or \(0 < k < 2/3\) only.
N.B. \(T = 0 \Rightarrow k = 2/3\) then answer is M0.
If they also solve \(T_C \geqslant 0\) or \(T_C > 0\), can still score M1 and possibly A1.