Find the modulus of elasticity of the string in terms of \(W\).
Find the angle that the force acting on the rod at \(A\) makes with the horizontal.
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A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(O P\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(Q O P = 90 ^ { \circ }\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac { 4 } { 5 }\).
Show that \(v ^ { 2 } = u ^ { 2 } + \frac { 14 } { 5 } a g\).
The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).
Obtain another equation relating \(u ^ { 2 } , v ^ { 2 } , a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\).
Find the least tension in the string during the motion.