CAIE M1 2014 June — Question 5

Exam BoardCAIE
ModuleM1 (Mechanics 1)
Year2014
SessionJune
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Mark schemeDownload PDF ↗
TopicPulley systems
TypeEnergy methods for pulley systems
DifficultyStandard +0.3 This is a standard M1 pulley system question requiring energy methods. Students must identify energy changes (KE gain, PE loss, work against friction) and apply conservation of energy. The friction coefficient and angle are given, making calculations straightforward. The multi-part structure guides students through the solution methodically, requiring only routine application of mechanics principles without novel insight.
Spec6.02d Mechanical energy: KE and PE concepts6.02e Calculate KE and PE: using formulae6.02i Conservation of energy: mechanical energy principle
5 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-3_343_691_254_725} A light inextensible rope has a block \(A\) of mass 5 kg attached at one end, and a block \(B\) of mass 16 kg attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Block \(A\) is held at rest at the bottom of the plane and block \(B\) hangs below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { \sqrt { 3 } }\). Block \(A\) is released from rest and the system starts to move. When each of the blocks has moved a distance of \(x \mathrm {~m}\) each has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down the gain in kinetic energy of the system in terms of \(v\).
  2. Find, in terms of \(x\),
    (a) the loss of gravitational potential energy of the system,
    (b) the work done against the frictional force.
  3. Show that \(21 v ^ { 2 } = 220 x\).
5\\
\includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-3_343_691_254_725}

A light inextensible rope has a block $A$ of mass 5 kg attached at one end, and a block $B$ of mass 16 kg attached at the other end. The rope passes over a smooth pulley which is fixed at the top of a rough plane inclined at an angle of $30 ^ { \circ }$ to the horizontal. Block $A$ is held at rest at the bottom of the plane and block $B$ hangs below the pulley (see diagram). The coefficient of friction between $A$ and the plane is $\frac { 1 } { \sqrt { 3 } }$. Block $A$ is released from rest and the system starts to move. When each of the blocks has moved a distance of $x \mathrm {~m}$ each has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(i) Write down the gain in kinetic energy of the system in terms of $v$.\\
(ii) Find, in terms of $x$,\\
(a) the loss of gravitational potential energy of the system,\\
(b) the work done against the frictional force.\\
(iii) Show that $21 v ^ { 2 } = 220 x$.

\hfill \mbox{\textit{CAIE M1 2014 Q5}}
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