| Exam Board | SPS |
|---|---|
| Module | SPS FM Mechanics (SPS FM Mechanics) |
| Year | 2025 |
| Session | January |
| Marks | 11 |
| Topic | Friction |
| Type | Elastic string with friction |
| Difficulty | Challenging +1.2 This is a multi-part mechanics problem requiring energy methods and equilibrium analysis with elastic strings and friction. While it involves several concepts (elastic potential energy, work done against friction, equilibrium conditions), the approach is relatively standard for Further Maths mechanics: part (a) uses work-energy principle with straightforward algebra, and part (b) applies static friction equilibrium. The constraint μ < tan θ guides the solution. More routine than problems requiring novel geometric insight or complex multi-stage reasoning. |
| Spec | 3.03v Motion on rough surface: including inclined planes6.02i Conservation of energy: mechanical energy principle |
4.
One end $A$ of a light elastic string $A B$, of modulus of elasticity $m g$ and natural length $a$, is fixed to a point on a rough plane inclined at an angle $\theta$ to the horizontal. The other end $B$ of the string is attached to a particle of mass $m$ which is held at rest on the plane. The string $A B$ lies along a line of greatest slope of the plane, with $B$ lower than $A$ and $A B = a$. The coefficient of friction between the particle and the plane is $\mu$, where $\mu < \tan \theta$. The particle is released from rest.
\begin{enumerate}[label=(\alph*)]
\item Show that when the particle comes to rest it has moved a distance $2 a ( \sin \theta - \mu \cos \theta )$ down the plane.
\item Given that there is no further motion, show that $\mu \geqslant \frac { 1 } { 3 } \tan \theta$.\\[0pt]
[Question 4 Continued]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Mechanics 2025 Q4 [11]}}