Year: 2008
Paper: 3
Question Number: 10
Course: UFM Mechanics
Section: Work, energy and Power 2
No solution available for this problem.
Most candidates attempted five, six or seven questions, and scored the majority of their total score on their best three or four. Those attempting seven or more tended not to do well, pursuing no single solution far enough to earn substantial marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
A long string consists of $n$ short
light strings joined together, each of natural length
$\ell$ and modulus of elasticity $\lambda$.
It hangs vertically at rest, suspended from one end.
Each of the short strings has a particle of mass $m$ attached to its lower
end. The short strings are numbered $1$ to $n$, the $n$th short
string being at the top. By considering the tension in the $r$th
short string, determine the length
of the long string. Find also the elastic energy stored in the long string.
A uniform heavy rope of mass $M$ and natural length $L_0$
has modulus of elasticity $\lambda$.
The rope hangs vertically at rest, suspended from one end.
Show that the length, $L$, of the rope is given by
\[
L=L_0\biggl(1+ \frac{Mg}{2\lambda}\biggr),
\]
and find an expression in terms of $L$, $L_0$ and $\lambda$ for the
elastic energy stored in the rope.
Just under a fifth attempted this, but many dealt successfully with the n short strings case to earn about half the marks. Occasionally a candidate would obtain the required length result for the heavy rope and fail to apply the same technique for the elastic energy, but apart from minor errors, most that appreciated how to take the limit had few difficulties.