Elastic string – horizontal circle on surface

5 One end of a light elastic string of natural length 0.3 m and modulus of elasticity 6 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg , which moves on the plane in a circular path with centre \(O\). The angular speed of \(P\) is \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. For the case \(\omega = 5\), calculate the extension of the string.
2. Express the extension of the string in terms of \(\omega\), and hence find the set of possible value of \(\omega\).

1 A particle \(P\) of mass 0.2 kg moves with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on a smooth surface. \(P\) is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to the point on the surface which is the centre of the circular motion of \(P\).

1. Find the radius of this circle.
2. Find the modulus of elasticity of the string.

5 A light elastic string has natural length \(a \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). When the length of the string is 1.6 m the tension is 4 N . When the length of the string is 2 m the tension is 6 N .

1. Find the values of \(a\) and \(\lambda\).
   One end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle \(P\) moves with constant speed on the surface in a circle with centre \(O\) and radius 1.9 m .
2. Find the speed of \(P\).

1 A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l \mathrm {~m}\). The speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the radius of the circle in which it moves is \(2 l \mathrm {~m}\). Calculate \(l\).

6 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 8 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves on the plane in a circular path with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the extension of the string is \(x \mathrm {~m}\).

1. Given that \(v = 2.5\), find \(x\).
   It is given instead that the kinetic energy of \(P\) is twice the elastic potential energy stored in the string.
2. Form two simultaneous equations and hence find \(x\) and \(v\).

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.

9 A particle, of mass 8 kg , is attached to one end of a length of elastic string. The particle is placed on a smooth horizontal surface. The other end of the elastic string is attached to a point \(O\) fixed on the horizontal surface. The elastic string has natural length 1.2 m and modulus of elasticity 192 N . \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-18_165_789_571_630} The particle is set in motion on the horizontal surface so that it moves in a circle, centre \(O\), with constant speed \(3 \mathrm {~ms} ^ { - 1 }\). Find the radius of the circle. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-19_2349_1691_221_153} \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-20_2505_1730_212_139}

A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l\) m. The speed of \(P\) is 4 m s\(^{-1}\), and the radius of the circle in which it moves is 2l m. Calculate \(l\). [4]

A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2m\) is attached to the free end of the string, this particle moves with speed \(\frac{1}{2}v\) in a horizontal circle with centre \(O\) and radius \(\frac{4}{3}x\).

1. Find \(x\) in terms of \(a\). [5]
2. Given that \(v = \sqrt{12ag}\), find the value of \(\lambda\). [2]

A particle of mass \(2\) kg is attached to one end of a light elastic string of natural length \(0.8\) m and modulus of elasticity \(100\) N. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is moving in a horizontal circle about \(O\) with the string taut and with constant angular speed \(5\) radians per second. Find the extension of the string. [3]