Year: 2012
Paper: 3
Question Number: 11
Course: UFM Mechanics
Section: Work, energy and Power 2
No solution available for this problem.
The number of candidates attempting more than six questions was, as last year, about 25%, though most of these extra attempts achieved little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
One end of a thin heavy uniform inextensible perfectly flexible rope of length $2L$
and mass $2M$
is attached to a fixed point $P$. A particle of mass $m$ is
attached to the other end. Initially, the particle is held at
$P$ and the rope hangs vertically in a loop below $P$. The particle is then released
so that it and a section of the rope (of decreasing length)
fall vertically as shown in the diagram.
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\psline[linewidth=0.1pt,linestyle=dashed,dash=2pt 2pt]{<->}(2.53,3.2)(2.53,5)
\psline(2.1,3.18)(2.06,0.25)
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\psline(2.02,0.26)(2.03,0)
\psline(2.03,0)(2.06,0.25)
\rput[tl](1.94,5.45){$P$}
\rput[tl](2.6,4.25){$x$}
\rput[tl](0.2,2.85){$L+\frac{1}{2}x$}
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You may assume that
each point on the moving section of the rope falls at the same speed as the
particle. Given that energy is conserved, show
that, when the particle has fallen a distance $x$ (where $x< 2L$),
its speed $v$ is given by
\[
v^2 = \frac { 2g x \big( mL +ML - \frac14 Mx)}{mL +ML - \frac12 Mx}\,.
\]
Hence show that the acceleration of the particle is
\[
g +
\frac{ Mgx\big(mL+ML- \frac14 Mx\big)}{2\big(mL +ML -\frac12 Mx\big)^2}\,
\,.\]
Deduce that the acceleration of the particle after it is
released is greater than $g$.
This was slightly less popular than question 10, and slightly less success was achieved. Most candidates correctly evaluated the kinetic and potential energies of the particle, and the kinetic energy of the rope. However they had more difficulty finding the potential energy of the rope, and put themselves at an unnecessary disadvantage by not explaining their logic. There were different ways of splitting up the rope, which one they used they frequently failed to make clear, and likewise those calculating potential energy relative to a reference point failed to make the choice of that point clear. The second part of the question was done very well using the result given for the first part. The last part was fairly easy, but quite a few candidates did not justify the logic fully.