2021 Paper 2 Q10

Year: 2021
Paper: 2
Question Number: 10

Course: UFM Mechanics
Section: Variable Force

Difficulty: 1500.0 Banger: 1500.0

non-inertial frame constrained motion energy method bead on wire constant acceleration parabolic constraint lagrangian mechanics

Problem

A train moves westwards on a straight horizontal track with constant acceleration \(a\), where \(a > 0\). Axes are chosen as follows: the origin is fixed in the train; the \(x\)-axis is in the direction of the track with the positive \(x\)-axis pointing to the East; and the positive \(y\)-axis points vertically upwards. A smooth wire is fixed in the train. It lies in the \(x\)--\(y\) plane and is bent in the shape given by \(ky = x^2\), where \(k\) is a positive constant. A small bead is threaded onto the wire. Initially, the bead is held at the origin. It is then released.

Explain why the bead cannot remain stationary relative to the train at the origin.
Show that, in the subsequent motion, the coordinates \((x, y)\) of the bead satisfy \[ \dot{x}(\ddot{x} - a) + \dot{y}(\ddot{y} + g) = 0 \] and deduce that \(\tfrac{1}{2}(\dot{x}^2 + \dot{y}^2) - ax + gy\) is constant during the motion.
Find an expression for the maximum vertical displacement, \(b\), of the bead from its initial position in terms of \(a\), \(k\) and \(g\).
Find the value of \(x\) for which the speed of the bead relative to the train is greatest and give this maximum speed in terms of \(a\), \(k\) and \(g\).

No solution available for this problem.

Examiner's report

— 2021 STEP 2, Question 10

This question was generally quite poorly attempted, with many candidates not able to understand fully the situation being studied. A large proportion of candidates only attempted the first part and were unable to earn any of the marks. Of the rest many did not progress beyond the second part, with many simply claiming incorrectly that the second derivative of x is a and the second derivative of y is −g. Those who did manage to solve the early parts of the question were generally quite successful with the rest of the question which was generally very well answered.

From the paper introduction

Candidates were generally well prepared for many of the questions on this paper, with the questions requiring more standard operations seeing the greatest levels of success. Candidates need to ensure that solutions to the questions are supported by sufficient evidence of the mathematical steps, for example when proving a given result or deducing the properties of graphs that are to be sketched. In a significant number of steps there were marks lost through simple errors such as mistakes in arithmetic or confusion of sine and cosine functions, so it is important for candidates to maintain accuracy in their solutions to these questions.

Source: Cambridge STEP 2021 Examiner's Report · 2021-p2.pdf

Rating Information

Difficulty Rating: 1500.0

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Banger Rating: 1500.0

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Problem source

A train moves westwards on a straight horizontal track with constant acceleration $a$, where $a > 0$. Axes are chosen as follows: the origin is fixed in the train; the $x$-axis is in the direction of the track with the positive $x$-axis pointing to the East; and the positive $y$-axis points vertically upwards.
 
A smooth wire is fixed in the train. It lies in the $x$--$y$ plane and is bent in the shape given by $ky = x^2$, where $k$ is a positive constant. A small bead is threaded onto the wire. Initially, the bead is held at the origin. It is then released.
 
\begin{questionparts}
    \item Explain why the bead cannot remain stationary relative to the train at the origin.
 
    \item Show that, in the subsequent motion, the coordinates $(x, y)$ of the bead satisfy
    \[
        \dot{x}(\ddot{x} - a) + \dot{y}(\ddot{y} + g) = 0
    \]
    and deduce that $\tfrac{1}{2}(\dot{x}^2 + \dot{y}^2) - ax + gy$ is constant during the motion.
 
    \item Find an expression for the maximum vertical displacement, $b$, of the bead from its initial position in terms of $a$, $k$ and $g$.
 
    \item Find the value of $x$ for which the speed of the bead relative to the train is greatest and give this maximum speed in terms of $a$, $k$ and $g$.
\end{questionparts}

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