| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2007 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Potential energy with inextensible strings or gravity only |
| Difficulty | Challenging +1.2 This is a multi-part mechanics question requiring center of mass calculations, potential energy formulation, equilibrium conditions via differentiation, and stability analysis via second derivative test. While it involves several steps and coordinate geometry with the right-angled framework, the techniques are standard M4 fare: finding PE from geometry, setting dV/dθ=0, and checking d²V/dθ². The geometric setup is clearly specified and the algebraic manipulation is straightforward once the framework is understood. This is moderately above average difficulty due to the multi-step nature and requiring careful geometric reasoning, but remains a typical examination question for this advanced mechanics module. |
| Spec | 6.02e Calculate KE and PE: using formulae6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks |
|---|---|
| \(V = -mga\cos\theta - mg(2a\cos\theta + a\sin\theta)\) | M1A1A1 |
| \(= -mga(3\cos\theta + \sin\theta)\) (+const) * | A1 |
| Total: 4 marks |
| Answer | Marks |
|---|---|
| \(\frac{dV}{d\theta} = -mga(-3\sin\theta + \cos\theta)\) | M1A1 |
| \(= 0 \Rightarrow \tan\theta = \frac{1}{3}\) | M1 |
| \(\Rightarrow \theta = 0.32(1)°\) or \(18.4°\) accept awrt | A1 |
| Total: 4 marks |
| Answer | Marks |
|---|---|
| \(\frac{d^2V}{d\theta^2} = -mga(-3\cos\theta - \sin\theta)\) | M1A1 |
| \(= mga(3\cos\theta + \sin\theta)\) | |
| \(\text{Hence, when } \theta = 0.32°, \frac{d^2V}{d\theta^2} > 0\) | M1 |
| i.e. stable | A1 |
| Total: 4 marks |
## Part (a)
| $V = -mga\cos\theta - mg(2a\cos\theta + a\sin\theta)$ | M1A1A1 |
| $= -mga(3\cos\theta + \sin\theta)$ (+const) * | A1 |
| **Total: 4 marks** | |
## Part (b)
| $\frac{dV}{d\theta} = -mga(-3\sin\theta + \cos\theta)$ | M1A1 |
| $= 0 \Rightarrow \tan\theta = \frac{1}{3}$ | M1 |
| $\Rightarrow \theta = 0.32(1)°$ or $18.4°$ accept awrt | A1 |
| **Total: 4 marks** | |
## Part (c)
| $\frac{d^2V}{d\theta^2} = -mga(-3\cos\theta - \sin\theta)$ | M1A1 |
| $= mga(3\cos\theta + \sin\theta)$ | |
| $\text{Hence, when } \theta = 0.32°, \frac{d^2V}{d\theta^2} > 0$ | M1 |
| i.e. stable | A1 |
| **Total: 4 marks** | |
### Guidance:
**a)** M1: Expression for the potential energy of the two rods. Condone trig errors. Condone sign errors. BC term in two parts. A1: correct expression for AB. A1: correct expression for BC. A1: Answer as given.
**b)** M1: Attempt to differentiate V. Condone errors in signs and in constants. A1: Derivative correct. M1: Set derivative = 0 and rearrange to a single trig function in $\theta$. A1: Solve for $\theta$. Or M1A1: find the position of the center of mass. M1A1: form and solve trig equation for $\theta$.
**c)** M1: Differentiate to obtain the second derivative. A1: Derivative correct. M1: Determine the sign of the second derivative. A1: Correct conclusion. cso. Or: M1: Find the value of $\frac{dV}{d\theta}$ on both sides of the minimum point. A1: signs correct. M1: Use the results to determine the nature of the turning point. A1: Correct conclusion, cso.
**These 4 marks are dependent on the use of derivatives.**
---\includegraphics{figure_1}
A framework consists of two uniform rods $AB$ and $BC$, each of mass $m$ and length $2a$, joined at $B$. The mid-points of the rods are joined by a light rod of length $a\sqrt{2}$, so that angle $ABC$ is a right angle. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis passes through the point $A$ and is perpendicular to the plane of the framework. The angle between the rod $AB$ and the downward vertical is denoted by $\theta$, as shown in Fig. 1.
\begin{enumerate}[label=(\alph*)]
\item Show that the potential energy of the framework is
$$-mga(3 \cos \theta + \sin \theta) + \text{constant}.$$ [4]
\item Find the value of $\theta$ when the framework is in equilibrium, with $B$ below the level of $A$. [4]
\item Determine the stability of this position of equilibrium. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2007 Q3 [12]}}