2000 Paper 2 Q11

Year: 2000
Paper: 2
Question Number: 11

Course: UFM Mechanics
Section: Simple Harmonic Motion

Difficulty: 1600.0 Banger: 1484.0

SHM elastic string Hooke's law energy conservation vertical motion modulus of elasticity period calculation free flight

Problem

The string $AP$ has a natural length of $1\!\cdot5\!$ metres and modulus of elasticity equal to $5g$ newtons. The end $A$ is attached to the ceiling of a room of height $2\!\cdot\!5$ metres and a particle of mass \mbox{$0\!\cdot\!5$ kg} is attached to the end $P$. The end $P$ is released from rest at a point $0\!\cdot\!5$ metres above the floor and vertically below $A$. Show that the string becomes slack, but that $P$ does not reach the ceiling. Show also that while the string is in tension, $P$ executes simple harmonic motion, and that the time in seconds that elapses from the instant when $P$ is released to the instant when $P$ first returns to its original position is $$ \left(\frac8{3g}\right)^{\!\frac12}+ \left(\frac3 {5g}\right)^{\!\frac12} {\Big(\pi - \arccos (3/7)\Big)}. $$ \noindent [Note that $\arccos x$ is another notation for $\cos^{-1} x$.]

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1484.0

Banger Comparisons: 1

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Problem source

The string $AP$ has a natural length of $1\!\cdot5\!$ metres and modulus of
elasticity equal to $5g$ newtons. The end $A$
is attached to the ceiling of a room of height $2\!\cdot\!5$ metres and 
a particle of mass \mbox{$0\!\cdot\!5$ kg}
is attached to the end $P$.  The  end  $P$  is 
released from  rest at a point $0\!\cdot\!5$ metres above the floor and vertically
below $A$.  Show that the string becomes slack, but that $P$ does not
reach the ceiling.
Show also that while the string is in tension, $P$ executes simple 
harmonic motion,
and that the time in seconds that elapses from the instant when $P$ is released
to the instant when $P$ first returns to its original position is
$$
\left(\frac8{3g}\right)^{\!\frac12}+
\left(\frac3 {5g}\right)^{\!\frac12}
{\Big(\pi - \arccos (3/7)\Big)}.
$$  
\noindent
[Note that $\arccos x$ is another notation for $\cos^{-1} x$.]

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