Work-energy with multiple stages

5 Each of two light elastic strings, \(S _ { 1 }\) and \(S _ { 2 }\), has modulus of elasticity 16 N . The string \(S _ { 1 }\) has natural length 0.4 m and the string \(S _ { 2 }\) has natural length 0.5 m . One end of \(S _ { 1 }\) is attached to a fixed point \(A\) of a smooth horizontal table and the other end is attached to a particle \(P\) of mass 0.5 kg . One end of \(S _ { 2 }\) is attached to a fixed point \(B\) of the table and the other end is attached to \(P\). The distance \(A B\) is 1.5 m . The particle \(P\) is held at \(A\) and then released from rest.

1. Find the speed of \(P\) at the instant that \(S _ { 2 }\) becomes slack.
2. Find the greatest distance of \(P\) from \(A\) in the subsequent motion.

7

1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
2. A block, of mass 4 kg , is attached to one end of a light elastic string. The string has natural length 2 m and modulus of elasticity 196 N . The other end of the string is attached to a fixed point \(O\).
   1. A second block, of mass 3 kg , is attached to the 4 kg block and the system hangs in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-16_374_291_890_877} Find the extension in the string.
   2. The block of mass 3 kg becomes detached from the 4 kg block and falls to the ground. The 4 kg block now begins to move vertically upwards. Find the extension of the string when the 4 kg block is next at rest.
   3. Find the extension of the string when the speed of the 4 kg block is a maximum.
      (3 marks)
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-18_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-19_2486_1714_221_153}

5 A light elastic string of natural length 1.6 m has modulus of elasticity 120 N . One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle \(P\) of weight 1.5 N . The particle is released from rest at the point \(A\), which is 2.1 m vertically below \(O\). It comes instantaneously to rest at \(B\), which is vertically above \(O\).

1. Verify that the distance \(A B\) is 4 m .
2. Find the maximum speed of \(P\) during its upward motion from \(A\) to \(B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_351_442_303_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_394_648_260_1018} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string of length \(0.8 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.4 kg and 0.58 kg respectively, attached to its ends. The string passes over a smooth horizontal cylinder of radius 0.8 m , which is fixed with its axis horizontal and passing through a fixed point \(O\). The string is held at rest in a vertical plane perpendicular to the axis of the cylinder, with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the cylinder through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).

6 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-4_638_473_260_836} A particle \(P\), of mass 3.5 kg , is in equilibrium suspended from the top \(A\) of a smooth slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 40 } { 49 }\), by an elastic rope of natural length 4 m and modulus of elasticity 112 N (see diagram). Another particle \(Q\), of mass 0.5 kg , is released from rest at \(A\) and slides freely downwards until it reaches \(P\) and becomes attached to it.

1. Find the value of \(V ^ { 2 }\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) immediately before it becomes attached to \(P\), and show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(\frac { 1 } { 2 } \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The combined particles slide downwards for a distance of \(X \mathrm {~m}\), before coming instantaneously to rest at \(B\).
2. Show that \(28 X ^ { 2 } - 8 X - 5 = 0\).