Year: 2024
Paper: 2
Question Number: 10
Course: LFM Pure and Mechanics
Section: Friction
No solution available for this problem.
Many candidates produced good solutions to the questions, with the majority of candidates opting to focus on the pure questions of the paper. Candidates demonstrated very good ability, particularly in the area of manipulating algebra. Many candidates produced clear diagrams which in many cases meant that they were more successful in their attempts at their questions than those who did not do so. The paper also contained a number of places where the answer to be reached was given in the question. In such cases, candidates must be careful to ensure that they provide sufficient evidence of the method used to reach the result in order to gain full credit.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A triangular prism lies on a horizontal plane. One of the rectangular faces of the prism is vertical; the second is horizontal and in contact with the plane; the third, oblique rectangular face makes an angle $\alpha$ with the horizontal. The two triangular faces of the prism are right angled triangles and are vertical. The prism has mass $M$ and it can move without friction across the plane.
A particle of mass $m$ lies on the oblique surface of the prism. The contact between the particle and the plane is rough, with coefficient of friction $\mu$.
\begin{questionparts}
\item Show that if $\mu < \tan\alpha$, then the system cannot be in equilibrium.
\end{questionparts}
Let $\mu = \tan\lambda$, with $0 < \lambda < \alpha < \frac{1}{4}\pi$.
A force $P$ is exerted on the vertical rectangular face of the prism, perpendicular to that face and directed towards the interior of the prism. The particle and prism accelerate, but the particle remains in the same position relative to the prism.
\begin{questionparts}
\setcounter{enumi}{1}
\item Show that the magnitude, $F$, of the frictional force between the particle and the prism is
\[ F = \frac{m}{M+m}\left|(M+m)g\sin\alpha - P\cos\alpha\right|. \]
Find a similar expression for the magnitude, $N$, of the normal reaction between the particle and the prism.
\item Hence show that the force $P$ must satisfy
\[ (M+m)g\tan(\alpha - \lambda) \leqslant P \leqslant (M+m)g\tan(\alpha + \lambda). \]
\end{questionparts}
This question received the fewest attempts, although a good proportion of the attempts that were made were successful. Many candidates drew clear diagrams in their attempts at this question. Part (i) was done fairly well, with most candidates resolving forces successfully. However, many candidates were not able to justify sufficiently well the situation in equilibrium as opposed to limiting equilibrium. In part (ii) many candidates struggled to know how to deal with the force that acts on the prism and thought instead that it would be acting on the particle. Those candidates who attempted part (iii) generally did well and many realised how to get both sides of the required inequality and were able to follow through the required manipulation.