Three non-collinear points \(A\), \(B\) and \(C\) lie in
a horizontal ceiling. A particle \(P\) of weight \(W\)
is suspended from this ceiling by means of three
light inextensible strings \(AP\), \(BP\) and \(CP\),
as shown in the diagram. The point \(O\) lies
vertically above \(P\) in the ceiling.
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The angles \(AOB\) and \(AOC\) are \(90^\circ+\theta\)
and \(90^\circ+\phi\), respectively, where \(\theta\) and \(\phi\)
are acute angles such that \(\tan\theta = \sqrt2\) and
\(\tan\phi =\frac14\sqrt2\).
The strings \(AP\), \(BP\) and \(CP\) make angles \(30^\circ\), \(90^\circ-\theta\)
and \(60^\circ\), respectively, with the vertical, and the tensions
in these strings have magnitudes \(T\), \(U\) and \(V\) respectively.
- Show that the unit vector in the direction \(PB\) can be written
in the form
\[
-\frac13\, {\bf i} - \frac{\sqrt2\,}3\, {\bf j} +
\frac{\sqrt2\, }{\sqrt3 \,} \,{\bf k}
\,,\]
where \(\bf i\,\), \(\, \bf j\) and \(\bf k\) are the usual mutually perpendicular
unit vectors
with \(\bf j\) parallel to \(OA\) and \(\bf k\) vertically upwards.
- Find expressions in vector form for the forces acting on \(P\).
- Show that \(U=\sqrt6 V\) and find \(T\), \(U\) and \(V\) in terms of \(W\).