| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2015 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic potential energy calculations |
| Difficulty | Challenging +1.2 This is a multi-part M4 question requiring energy methods and equilibrium analysis. Part (a) involves careful geometry and setting up PE expressions (moderately challenging setup), part (b) requires differentiation and solving a trigonometric equation (standard technique), and part (c) tests second derivative for stability (routine). The geometric setup with the semicircular wire and hanging masses requires careful visualization, but the mathematical techniques are standard for M4. Slightly above average difficulty due to the geometric complexity and multi-step nature, but follows predictable M4 patterns. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives6.02e Calculate KE and PE: using formulae6.04e Rigid body equilibrium: coplanar forces |
A smooth wire, with ends A and B, is in the shape of a semicircle of radius $r$. The line AB is horizontal and the midpoint of AB is O. The wire is fixed in a vertical plane. A small ring R of mass $2m$ is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at A and is attached to a particle of mass $6m$. The other string passes through a small smooth ring fixed at B and is attached to a second particle of mass $6m$. The particles hang freely under gravity. The angle between the radius OR and the downward vertical is $2\theta$, where $-\frac{\pi}{4} < \theta < \frac{\pi}{4}$
(a) Show that the potential energy of the system is
$$2mgr(3\cos\theta - \cos 2\theta) + \text{constant}$$
(6 marks)
(b) Find the values of $\theta$ for which the system is in equilibrium.
(4 marks)
(c) Determine the stability of the position of equilibrium for which $\theta > 0$
(3 marks)
---6.
A smooth wire, with ends $A$ and $B$, is in the shape of a semicircle of radius $r$. The line $A B$ is horizontal and the midpoint of $A B$ is $O$. The wire is fixed in a vertical plane. A small ring $R$ of mass $2 m$ is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at $A$ and is attached to a particle of mass $\sqrt { 6 } m$. The other string passes through a small smooth ring fixed at $B$ and is attached to a second particle of mass $\sqrt { 6 } \mathrm {~m}$. The particles hang freely under gravity, as shown in Figure 3. The angle between the radius $O R$ and the downward vertical is $2 \theta$, where $- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }$
\begin{enumerate}[label=(\alph*)]
\item Show that the potential energy of the system is
$$2 m g r ( 2 \sqrt { 3 } \cos \theta - \cos 2 \theta ) + \text { constant }$$
\item Find the values of $\theta$ for which the system is in equilibrium.
\item Determine the stability of the position of equilibrium for which $\theta > 0$
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2015 Q6 [13]}}