8 One end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\) is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\) which is on a horizontal surface. The surface is modelled as being smooth and \(A\) moves in a circular path around \(O\) with constant speed \(v \mathrm {~ms} ^ { - 1 }\). The extension of the string is denoted by \(x \mathrm {~m}\).

1. Show that \(x\) satisfies \(\lambda x ^ { 2 } + \lambda | x - | m v ^ { 2 } = 0\).
2. By solving the equation in part (a) and using a binomial series, show that if \(\lambda\) is very large then \(\lambda \mathrm { x } \approx \mathrm { mv } ^ { 2 }\).
3. By considering the tension in the string, explain how the result obtained when \(\lambda\) is very large relates to the situation when the string is inextensible. The nature of the horizontal surface is such that the modelling assumption that it is smooth is justifiable provided that the speed of the particle does not exceed \(7 \mathrm {~ms} ^ { - 1 }\). In the case where \(m = 0.16\) and \(\lambda = 260\), the extension of the string is measured as being 3.0 cm .
4. Estimate the value of \(v\).
5. Explain whether the value of \(v\) means that the modelling assumption is necessarily justifiable in this situation.