Coalescing particles collision

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(2 m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). Initially the particle is at the point \(A\) where \(O A = a\) and \(O A\) makes an angle \(60 ^ { \circ }\) with the downward vertical. The particle is projected downwards from \(A\) with speed \(u\) in a direction perpendicular to the string, as shown in Figure 3. The point \(B\) is vertically below \(O\) and \(O B = a\). As \(P\) passes through \(B\) it strikes and adheres to another particle \(Q\) of mass \(m\) which is at rest at \(B\).

1. Show that the speed of the combined particle immediately after the impact is $$\frac { 2 } { 3 } \sqrt { u ^ { 2 } + a g } .$$
2. Find, in terms of \(a , g , m\) and \(u\), the tension in the string immediately after the impact. The combined particle moves in a complete circle.
3. Show that \(u ^ { 2 } \geqslant \frac { 41 a g } { 4 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and horizontal. It is projected downwards with speed \(\sqrt { } ( 12 a g )\). At the lowest point of its motion, \(P\) collides directly with a particle \(Q\) of mass \(k m\) which is at rest (see diagram). In the collision, \(P\) and \(Q\) coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of \(k\).

4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).

5 A particle \(P\), of mass 2.5 kg , is in equilibrium suspended from a fixed point \(A\) by a light elastic string of natural length 3 m and modulus of elasticity 36.75 N . Another particle \(Q\), of mass 1 kg , is released from rest at \(A\) and falls freely until it reaches \(P\) and becomes attached to it.

1. Show that the speed of the combined particles, immediately after \(Q\) becomes attached to \(P\), is \(2 \sqrt { 2 } \mathrm {~ms} ^ { - 1 }\). The combined particles fall a further distance \(X \mathrm {~m}\) before coming to instantaneous rest.
2. Find a quadratic equation satisfied by \(X\), and show that it simplifies to \(35 X ^ { 2 } - 56 X - 80 = 0\).

3. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate about a fixed smooth axis which passes through \(A\) and is perpendicular to the rod. The rod has angular speed \(\omega\) when it strikes a particle \(P\) of mass \(m\) and adheres to it. Immediately before the rod strikes \(P , P\) is at rest and at a distance \(x\) from \(A\). Immediately after the rod strikes \(P\), the angular speed of the rod is \(\frac { 3 } { 4 } \omega\). Find \(x\) in terms of \(a\).
(5)

3. A uniform rectangular lamina \(A B C D\), where \(A B = a\) and \(B C = 2 a\), has mass \(2 m\). The lamina is free to rotate about its edge \(A B\), which is fixed and vertical. The lamina is at rest when it is struck at \(C\) by a particle \(P\) of mass \(m\). The particle \(P\) is moving horizontally with speed \(U\) in a direction which is perpendicular to the lamina. The coefficient of restitution between \(P\) and the lamina is 0.5 Find the angular speed of the lamina immediately after the impact.
(8)

2. A rod \(A B\) has mass \(m\) and length \(4 a\). It is free to rotate about a fixed smooth horizontal axis through the point \(O\) of the rod, where \(A O = a\). The rod is hanging in equilibrium with \(B\) below \(O\) when it is struck by a particle \(P\), of mass \(3 m\), moving horizontally with speed \(v\). When \(P\) strikes the rod, it adheres to it. Immediately after striking the rod, \(P\) has speed \(\frac { 2 } { 3 } v\). Find the distance from \(O\) of the point where \(P\) strikes the rod.
(7 marks)

14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
5

1. Show that the distance of the centre of mass of the solid from the circular face is \(\frac { 5 } { 8 }\) [0pt] [5 marks]
   5
2. The solid is suspended in equilibrium from a point on the edge of the circular face.
   Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
   6 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
   The other end of the string is attached to a fixed point \(O\) The sphere is released from rest with the string taut and at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
   1. Find the speed of the sphere when it is directly below \(O\) 6
   2. State one assumption that you made about the string.
      6
   3. As the sphere moves, the string makes an angle \(\theta\) with the downward vertical. By finding an expression for the tension in the string in terms of \(\theta\), show that the tension is a maximum when the sphere is directly below \(O\) 6
   4. A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below \(O\) They find that the tension is 9.5 newtons.
      Explain why this result is reasonable, showing any calculations that you make.
      7 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. A straight line has been marked on the surface and the particles are on opposite sides of the line. Particle \(A\) has mass 2 kg and moves with velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line. Particle \(B\) has mass 3 kg and moves with velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) to the line. The particles and their velocities are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-08_451_739_858_653} The particles collide when they reach the line and then move together as a single combined particle. 7
   5. Show that the magnitude of the impulse on particle \(A\) during the collision is 7.55 Ns correct to three significant figures.
      7
   6. State the magnitude of the impulse on \(B\) during the collision, giving a reason for your answer. 7
   7. Find the size of the angle between the straight line and the impulse acting on \(B\), giving your answer to the nearest degree. 7
   8. During the collision, one particle crosses the straight line.
      State which particle crosses the line, giving a reason for your answer.
      [0pt] [1 mark] 8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A block has mass 10 kg and is at rest 1 metre from a fixed point \(O\) on a horizontal surface. One end of an elastic string is attached to the block and the other end of the elastic string is attached to the point \(O\) The elastic string has modulus of elasticity 40 newtons and natural length 2 metres.
      The coefficient of friction between the block and the surface is 0.6 A force is applied to the block so that it starts to move towards a vertical wall.
      The block moves on a line that is perpendicular to the wall.
      The force has magnitude 100 newtons and acts at an angle of \(30 ^ { \circ }\) to the horizontal.
      The situation is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-10_239_1339_1176_354} 8
   9. Show that the distance that the block has moved, when the forces acting on it are in equilibrium, is 3.9 metres correct to two significant figures.
      [0pt] [4 marks]
      8
   10. State one limitation of the model that you have used. 8
   11. Find the maximum speed of the block.
       8
   12. The vertical wall is 8.7 metres from \(O\) Determine whether the block reaches the wall, showing any calculations that you make. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-13_2492_1721_217_150}