Question 13 - A-Level Maths

OCR H240/03 2017 Specimen — Question 13 8 marks

Exam Board	OCR
Module	H240/03 (Pure Mathematics and Mechanics)
Year	2017
Session	Specimen
Marks	8
Topic	Pulley systems
Type	Particle on smooth incline connected to hanging particle
Difficulty	Standard +0.3 This is a standard M1 connected particles problem with friction on one plane. Part (a) requires resolving forces, applying F=ma to both particles, and solving simultaneous equations—routine techniques with straightforward arithmetic. Part (b) is a simple application of v²=u²+2as. The given angle simplifies to sin=4/5, cos=3/5, making calculations clean. Slightly above average due to the two-plane setup and friction, but still a textbook exercise requiring no novel insight.
Spec	3.02d Constant acceleration: SUVAT formulae 3.03o Advanced connected particles: and pulleys 3.03v Motion on rough surface: including inclined planes

Particle \(A\), of mass \(m\) kg, lies on the plane \(\Pi_1\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. Particle \(B\), of \(4m\) kg, lies on the plane \(\Pi_2\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\). The coefficient of friction between particle \(A\) and \(\Pi_1\) is \(\frac{1}{4}\) and plane \(\Pi_2\) is smooth. Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below. \includegraphics{figure_13}

Show that when \(A\) is released it accelerates towards the pulley at \(\frac{7g}{15}\) m s\(^{-2}\). [6]
Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac{1}{4}\) m when its speed is \(\sqrt{\frac{7g}{30}}\) m s\(^{-1}\). [2]

Show LaTeX source

Particle $A$, of mass $m$ kg, lies on the plane $\Pi_1$ inclined at an angle of $\tan^{-1} \frac{4}{3}$ to the horizontal.

Particle $B$, of $4m$ kg, lies on the plane $\Pi_2$ inclined at an angle of $\tan^{-1} \frac{4}{3}$ to the horizontal.

The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at $P$. The coefficient of friction between particle $A$ and $\Pi_1$ is $\frac{1}{4}$ and plane $\Pi_2$ is smooth.

Particle $A$ is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane.

This is shown on the diagram below.

\includegraphics{figure_13}

\begin{enumerate}[label=(\alph*)]
\item Show that when $A$ is released it accelerates towards the pulley at $\frac{7g}{15}$ m s$^{-2}$. [6]

\item Assuming that $A$ does not reach the pulley, show that it has moved a distance of $\frac{1}{4}$ m when its speed is $\sqrt{\frac{7g}{30}}$ m s$^{-1}$. [2]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03 2017 Q13 [8]}}

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