Horizontal elastic string on smooth surface

2 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 64 N . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal surface. \(P\) is placed on the surface at a point 0.8 m from \(A\). The particle \(P\) is then projected with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly away from \(A\).

1. Calculate the distance \(A P\) when \(P\) is at instantaneous rest.
2. Calculate the speed of \(P\) when it is 1.0 m from \(A\).

1 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is projected horizontally from \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the greatest distance of the particle from \(O\).

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

6 An elastic string has one end attached to a point \(O\), fixed on a horizontal table. The other end of the string is attached to a particle of mass 5 kilograms. The elastic string has natural length 2 metres and modulus of elasticity 200 newtons. The particle is pulled so that it is 2.5 metres from the point \(O\) and it is then released from rest on the table.

1. Calculate the elastic potential energy when the particle is 2.5 m from the point \(O\).
2. If the table is smooth, show that the speed of the particle when the string becomes slack is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The table is, in fact, rough and the coefficient of friction between the particle and the table is 0.4 . Find the speed of the particle when the string becomes slack.

9 At a theme park, a light elastic rope is used to bring a carriage to rest at the end of a ride. The carriage has mass 200 kg and is travelling at \(8 \mathrm {~ms} ^ { - 1 }\) when the elastic rope is attached to the carriage as it passes over a point \(O\). The other end of the elastic rope is fixed to the point \(O\). The carriage then moves along a horizontal surface until it is brought to rest. The elastic rope is then detached so that the carriage remains at rest. The elastic rope has natural length 6 m and modulus of elasticity 1800 N . The rope, once taut, remains horizontal throughout the motion.

1. Calculate the elastic potential energy of the rope when the carriage is 10 m from \(O\).
   (3 marks)
2. A student's simple model assumes that there are no resistance forces acting on the carriage so that it is brought to rest by the elastic rope alone. Find the distance of the carriage from \(O\) when it is brought to rest.
3. The student improves the model by also including a constant resistance force of 800 N which acts while the carriage is in motion. Find the distance of the carriage from \(O\) when it is brought to rest.
   (8 marks)

8 Zoë carries out an experiment with a block, which she places on the horizontal surface of an ice rink. She attaches one end of a light elastic string to a fixed point, \(A\), on a vertical wall at the edge of the ice rink at the height of the surface of the ice rink. The block, of mass 0.4 kg , is attached to the other end of the string. The string has natural length 5 m and modulus of elasticity 120 N . The block is modelled as a particle which is placed on the surface of the ice rink at a point \(B\), where \(A B\) is perpendicular to the wall and of length 5.5 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-6_499_1429_813_333} The block is set into motion at the point \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) directly towards the point \(A\). The string remains horizontal throughout the motion.

1. Initially, Zoë assumes that the surface of the ice rink is smooth. Using this assumption, find the speed of the block when it reaches the point \(A\).
2. Zoë now assumes that friction acts on the block. The coefficient of friction between the block and the surface of the ice rink is \(\mu\).
   1. Find, in terms of \(g\) and \(\mu\), the speed of the block when it reaches the point \(A\).
   2. The block rebounds from the wall in the direction of the point \(B\). The speed of the block immediately after the rebound is half of the speed with which it hit the wall. Find \(\mu\) if the block comes to rest just as it reaches the point \(B\).

1 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.8 m and modulus of elasticity 1.35 mg N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. \(P\) is held at rest at a point on the surface 3 m from \(O\). The particle is then released. Find

1. the initial acceleration of \(P\),
2. the speed of \(P\) at the instant the string becomes slack.

One end of a light elastic string of natural length \(0.7\) m is attached to a fixed point \(A\) on a smooth horizontal surface. The other end of the string is attached to a particle \(P\) of mass \(0.3\) kg which is held at a point \(B\) on the horizontal surface, where \(AB = 1.2\) m. It is given that \(P\) is released from rest at \(B\) and that when \(AP = 0.9\) m, the particle has speed \(4\) m s\(^{-1}\). Calculate the modulus of elasticity of the string. [3]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda mg\), is attached to a fixed point \(O\). The string lies on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected in the direction \(OP\). When the length of the string is \(\frac{4}{3}a\), the speed of \(P\) is \(\sqrt{2ag}\). When the length of the string is \(\frac{5}{3}a\), the speed of \(P\) is \(\frac{1}{2}\sqrt{2ag}\). Find the value of \(\lambda\). [4]

A light elastic string \(AB\) of natural length 1.5 m has modulus of elasticity 20 N. The end \(A\) is fixed to a point on a smooth horizontal table. A small ball \(S\) of mass 0.2 kg is attached to the end \(B\). Initially \(S\) is at rest on the table with \(AB = 1.5\) m. The ball \(S\) is then projected horizontally directly away from \(A\) with a speed of 5 m s\(^{-1}\). By modelling \(S\) as a particle,

1. find the speed of \(S\) when \(AS = 2\) m. [5]

When the speed of \(S\) is 1.5 m s\(^{-1}\), the string breaks.

1. Find the tension in the string immediately before the string breaks. [5]

A particle of mass \(m\) kg is attached to one end of a light elastic string of natural length \(a\) metres and modulus of elasticity \(5ma\) newtons. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length \(2a\) metres and then released at time \(t = 0\). During the subsequent motion, when the particle is moving with speed \(v\) m s\(^{-1}\), the particle experiences a resistance of magnitude \(4mv\) newtons. At time \(t\) seconds after the particle is released, the length of the string is \((a + x)\) metres, where \(0 \leqslant x \leqslant a\).

1. Show that, from \(t = 0\) until the string becomes slack, $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0$$ [3]
2. Hence express \(x\) in terms of \(a\) and \(t\). [6]
3. Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form \(ka\), where \(k\) is a constant to be found correct to 2 significant figures. [4]