Year: 2022
Paper: 2
Question Number: 9
Course: UFM Mechanics
Section: Moments
No solution available for this problem.
Candidates appeared to be generally well prepared for most topics within the examination, but there were a few situations in questions where some did not appear to be as proficient in standard techniques as needed. In particular, the method for finding invariant lines required in question 8 and the manipulation of trigonometric functions that were needed in question 10 caused considerable difficulties for some candidates. An additional issue that occurred at numerous points in the paper relates to the direction in which a deduction is required. It is important that candidates make sure that they know which statement is the one that they should start from as they deduce the other and that it is clear in their solution that the logic has gone in the correct direction. Clarity of solution is also an issue that candidates should be aware of, especially in the situations where the result to be reached has been given. It is important to check that there are no special cases that need to be considered separately, and when dividing by a function it is necessary to confirm that the function cannot be equal to 0 (and in the case of inequalities that the function always has the same sign). When drawing diagrams and sketching graphs it is useful if significant points that need to be clear are not drawn over the lines on the page as these can be difficult to interpret during the marking process.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A rectangular prism is fixed on a horizontal surface. A vertical wall, parallel to a vertical face of the prism, stands at a distance $d$ from it. A light plank, making an acute angle $\theta$ with the horizontal, rests on an upper edge of the prism and is in contact with the wall below the level of that edge of the prism and above the level of the horizontal plane. You may assume that the plank is long enough and the prism high enough to make this possible.
The contact between the plank and the prism is smooth, and the coefficient of friction at the contact between the plank and the wall is $\mu$. When a heavy point mass is fixed to the plank at a distance $x$, along the plank, from its point of contact with the wall, the system is in equilibrium.
\begin{questionparts}
\item Show that, if $x = d\sec^3\theta$, then there is no frictional force acting between the plank and the wall.
\item Show that, if $x > d\sec^3\theta$, it is necessary that
\[\mu \geqslant \frac{x - d\sec^3\theta}{x\tan\theta}\]
and give the corresponding inequality if $x < d\sec^3\theta$.
\item Show that
\[\frac{x}{d} \geqslant \frac{\sec^3\theta}{1 + \mu\tan\theta}\,.\]
Show also that, if $\mu < \cot\theta$, then
\[\frac{x}{d} \leqslant \frac{\sec^3\theta}{1 - \mu\tan\theta}\,.\]
\item Show that if $x$ is such that the point mass is fixed to the plank somewhere between the edge of the prism and the wall, then $\tan\theta < \mu$.
\end{questionparts}
Many of the attempts at this question failed to achieve good marks. Some candidates struggled to convert the text into a suitable diagram, while others were confused about whether to label the forces that other objects exerted on the plank or the forces exerted by the plank itself. These diagrams then often led to confusion about the directions of the forces in play. In part (i) there were many attempts that used backwards logic – assuming that there was no frictional force and finding the value of x required for this to be true. There were also a number of candidates who assumed that the frictional force would always be equal to μR, rather than taking this as the maximum. Part (ii) was often done poorly because it was often not clear that the solution was using the magnitudes of the forces when using F = μR. In part (iii) candidates often failed to consider all possible cases for the value of x and it was common to see the first result shown in one of the cases and the second result shown in the other. Solutions to part (iv) were more successful from the candidates who reached this point.