In the figure, \(W_{1}\) and \(W_{2}\) are wheels, both of radius \(r\).
Their centres \(C_{1}\) and \(C_{2}\) are fixed at the same height,
a distance \(d\) apart, and each wheel is free to rotate, without friction,
about its centre. Both wheels are in the same vertical plane. Particles
of mass \(m\) are suspended from \(W_{1}\) and \(W_{2}\) as shown, by
light inextensible strings would round the wheels. A light elastic
string of natural length \(d\) and modulus elasticity \(\lambda\) is
fixed to the rims of the wheels at the points \(P_{1}\) and \(P_{2}.\)
The lines joining \(C_{1}\) to \(P_{1}\) and \(C_{2}\) to \(P_{2}\) both
make an angle \(\theta\) with the vertical. The system is in equilibrium.
\noindent
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\par
\vspace{-0.5cm}
Show that
\[ \sin2\theta=\frac{mgd}{\lambda r}.
\]For what value or values of
\(\lambda\) (in terms of \(m,d,r\) and \(g\)) are there
- sep}{3mm}
- \(\bf (i)\) no equilibrium positions,
- \(\bf (ii)\) just one equilibrium position,
- \(\bf (iii)\) exactly two equilibrium positions,
- \(\bf (iv)\) more than two equilibrium positions?