M4 June 2017 Q7
7.
Figure 2 shows four uniform rods, each of mass \(m\) and length \(2a\). The rods are freely hinged at their ends to form a rhombus \(ABCD\). Point \(A\) is attached to a fixed point on a ceiling and the rhombus hangs freely with \(C\) vertically below \(A\). A light elastic spring of natural length \(2a\) and modulus of elasticity \(7mg\) connects the points \(A\) and \(C\). A particle of mass \(3m\) is attached to point \(C\).
(a) Show that, when \(AD\) is at an angle \(\theta\) to the downward vertical, the potential energy \(V\) of the system is given by \[V = 28mga\cos^2\theta - 48mga\cos\theta + \text{constant}\] (5)
Given that \(\theta \gt 0\)
(b) find the value of \(\theta\) for which the system is in equilibrium, (4)
(c) determine the stability of this position of equilibrium. (4)
| Scheme | Marks |
|---|---|
| Relative to the fixed point A, PE of mass at C \(= -3mg\times 4a\cos\theta\) | B1 |
| PE of rods \(= -2mg\times a\cos\theta - 2mg\times 3a\cos\theta\) | B1 |
| Extension in the spring \(= 4a\cos\theta - 2a\) | B1 |
| \(\dfrac{7mg(4a\cos\theta - 2a)^2}{4a} - 20mga\cos\theta\) | M1 |
| \(= 7mga\left(4\cos^2\theta - 4\cos\theta + 1\right) - 20mga\cos\theta\) | |
| \(= 28mga\cos^2\theta - 48mga\cos\theta + \text{constant}\) | A1 |
| (5) |
Notes
M1 Total PE
A1 *given answer*
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}V}{\mathrm{d}\theta} =\) | M1 |
| \(= -56mga\cos\theta\sin\theta + 48mga\sin\theta\) | A1 |
| \(8\sin\theta(-7\cos\theta + 6) = 0\) | M1 |
| \(\Rightarrow \theta = \cos^{-1}\dfrac{6}{7}\ (31^\circ, 0.54\text{r})\) | A1 |
| (4) |
Notes
M1 Differentiate
M1 \(\dfrac{\mathrm{d}V}{\mathrm{d}\theta} = 0\) and solve for \(\theta\)
| Scheme | Marks |
|---|---|
| M1 | |
| \(\dfrac{\mathrm{d}^2V}{\mathrm{d}\theta^2} = -56mga\left(\cos^2\theta - \sin^2\theta\right) + 48mga\cos\theta\) | A1 |
| \(= -56mga\left(2\times\dfrac{36}{49} - 1\right) + 48mga\times\dfrac{6}{7}\) | DM1 |
| \(= \dfrac{728}{49}mga \gt 0,\qquad\) stable | A1 |
| (4) | |
| (13 marks) |
Notes
M1 Differentiate to obtain second derivative
DM1 Find value of second derivative when \(\theta = \cos^{-1}\dfrac{6}{7}\). Dependent on preceding M1
A1 \((14.8\ldots mga)\)