Elastic string vertical motion

8 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-4_933_275_689_934} The diagram shows a light elastic string of natural length 0.6 m and modulus of elasticity 5 N with one end attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at the point \(A\), which is 0.9 m vertically above \(O\). The particle is released from rest and travels vertically downwards through \(O\) to the point \(C\), where it starts to move upwards. \(B\) is the point of the line \(A C\) where the string first becomes slack.

1. Find the speed of \(P\) at \(B\).
2. The extension of the string when \(P\) is at \(C\) is \(x \mathrm {~m}\).
   1. Show that \(x ^ { 2 } - 0.48 x - 0.81 = 0\).
   2. Hence find the distance \(A C\).

7 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 32 N . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest. A horizontal plane is fixed at a distance 1 m below \(O\). The particle \(P\) is again released from rest at \(O\).
2. Calculate the speed of \(P\) immediately before it collides with the plane.
3. In the collision with the plane, \(P\) loses \(96 \%\) of its kinetic energy. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest above the plane, given that this occurs when the string is slack.

1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\).

One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(\frac { 1 } { 2 } m\) is attached to the other end of the elastic string. The point \(A\) is vertically below \(O\) with \(O A = 4 a\). Particle \(P\) is held at \(A\) and released from rest. The speed of \(P\) at the instant when it has moved a distance \(a\) upwards is \(\sqrt { 3 a g }\) Air resistance to the motion of \(P\) is modelled as having magnitude \(k m g\), where \(k\) is a constant. Using the model and the work-energy principle,

1. show that \(k = \frac { 1 } { 4 }\) Particle \(P\) is now held at \(O\) and released from rest. As \(P\) moves downwards, it reaches its maximum speed as it passes through the point \(B\).
2. Find the distance \(O B\).

3 A light elastic string has natural length 1.2 m and stiffness \(637 \mathrm { Nm } ^ { - 1 }\).

1. The string is stretched to a length of 1.3 m . Find the tension in the string and the elastic energy stored in the string. One end of this string is attached to a fixed point \(A\). The other end is attached to a heavy ring \(R\) which is free to move along a smooth vertical wire. The shortest distance from A to the wire is 1.2 m (see Fig. 3). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-4_357_337_669_863} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The ring is in equilibrium when the length of the string \(A R\) is 1.3 m .
2. Show that the mass of the ring is 2.5 kg . The ring is given an initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from its equilibrium position. It first comes to rest, instantaneously, in the position where the length of AR is 1.5 m .
3. Find \(u\).
4. Determine whether the ring will rise above the level of A .