Challenging +1.8 This is a challenging M2 mechanics problem requiring energy methods with elastic strings, friction on an incline, and careful consideration of direction changes. Students must determine which direction P moves initially, apply work-energy principles accounting for variable elastic forces and friction (which changes direction), and solve for the final position. The multi-stage nature (checking initial motion, applying energy conservation with friction doing negative work throughout) and the need to handle two elastic strings simultaneously make this significantly harder than standard M2 questions, though the individual techniques are syllabus-standard.
A particle \(P\), of mass \(5\) kg is placed at the point \(A\) on a rough plane which is inclined at \(30°\) to the horizontal.
The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(QR = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(AQ = 4\) metres and \(AR = 11\) metres.
The three points \(Q\), \(A\) and \(R\) are on a line of greatest slope of the plane.
\includegraphics{figure_8}
The particle is attached to two light elastic strings, \(PQ\) and \(PR\).
One of the strings, \(PQ\), has natural length \(4\) metres and modulus of elasticity \(160\) N, the other string, \(PR\), has natural length \(6\) metres and modulus of elasticity \(120\) N.
The particle is released from rest at the point \(A\).
The coefficient of friction between the particle and the plane is \(0.4\).
Find the distance of the particle from \(Q\) when it is next at rest.
[8 marks]
A particle $P$, of mass $5$ kg is placed at the point $A$ on a rough plane which is inclined at $30°$ to the horizontal.
The points $Q$ and $R$ are also on the surface of the inclined plane, with $QR = 15$ metres. The point $A$ is between $Q$ and $R$ so that $AQ = 4$ metres and $AR = 11$ metres.
The three points $Q$, $A$ and $R$ are on a line of greatest slope of the plane.
\includegraphics{figure_8}
The particle is attached to two light elastic strings, $PQ$ and $PR$.
One of the strings, $PQ$, has natural length $4$ metres and modulus of elasticity $160$ N, the other string, $PR$, has natural length $6$ metres and modulus of elasticity $120$ N.
The particle is released from rest at the point $A$.
The coefficient of friction between the particle and the plane is $0.4$.
Find the distance of the particle from $Q$ when it is next at rest.
[8 marks]
\hfill \mbox{\textit{AQA M2 2016 Q8 [8]}}