Year: 2023
Paper: 2
Question Number: 9
Course: LFM Pure and Mechanics
Section: Newton's laws and connected particles
No solution available for this problem.
Many candidates were able to express their reasoning clearly and presented good solutions to the questions that they attempted. There were excellent solutions seen for all of the questions. An area where candidates struggled in several questions was in the direction of the logic that was required in a solution. Some candidates failed to appreciate that separate arguments may be needed for the "if" and "only if" parts of a question and, in some cases, candidates produced correct arguments, but for the wrong direction. In several questions it was clear that candidates who used sketches or diagrams generally performed much better that those who did not. Sketches often also helped to make the solution clearer and easier to understand. Several questions on the STEP papers ask candidates to show a given result. Candidates should be aware that there is a need to present sufficient detail in their solutions so that it is clear that the reasoning is well understood.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A truck of mass $M$ is connected by a light, rigid tow-bar, which is parallel to the ground, to a trailer of mass $kM$. A constant driving force $D$ which is parallel to the ground acts on the truck, and the only resistance to motion is a frictional force acting on the trailer, with coefficient of friction $\mu$.
\begin{itemize}
\item When the truck pulls the trailer up a slope which makes an angle $\alpha$ to the horizontal, the acceleration is $a_1$ and there is a tension $T_1$ in the tow-bar.
\item When the truck pulls the trailer on horizontal ground, the acceleration is $a_2$ and there is a tension $T_2$ in the tow-bar.
\item When the truck pulls the trailer down a slope which makes an angle $\alpha$ to the horizontal, the acceleration is $a_3$ and there is a tension $T_3$ in the tow-bar.
\end{itemize}
All accelerations are taken to be positive when in the direction of motion of the truck.
\begin{questionparts}
\item Show that $T_1 = T_3$ and that $M(a_1 + a_3 - 2a_2) = 2(T_2 - T_1)$.
\item It is given that $\mu < 1$.
\begin{enumerate}
\item[(a)] Show that
\[a_2 < \tfrac{1}{2}(a_1 + a_3) < a_3\,.\]
\item[(b)] Show further that
\[a_1 < a_2\,.\]
\end{enumerate}
\end{questionparts}
Less than half of the candidates produced an accurate diagram for this question, with many leaving off some forces, or making errors with the gravitational force by not including g. This had an impact on their ability to proceed with the question, and often those with poorly presented diagrams had sign errors in their force balance equations (for example, with tension in the wrong direction). Most seemed to understand how to calculate frictional force. The inclusion of friction on the trailer and not the truck clearly confused some candidates, causing many of the question parts to be inaccessible. Many candidates seemed to struggle with the fact that 6 equations had to be dealt with, and so struggled to identify which variables to eliminate and how to eliminate them. Additionally, some candidates did not realise that some of the forces would take different values in the different cases being considered. Part (i) was done quite well overall, although for the second part, a fair number of candidates showed each side was equal to some expression involving the other variables, which is valid but took much more time than the direction approach using the equation of motion for the truck. Part (ii)(a) this part was done well in some cases, although less well than the previous part. Most candidates who attempted this part were able to show the upper inequality, but the lower one proved to be more difficult. Most attempts to part (ii)(b) only achieved two of the marks available. Many candidates did not recognise that the half angle formula was useful here and so struggled to make progress on the question.