Collision/impulse during circular motion

A question is this type if and only if a particle in circular motion experiences a collision or impulse that changes its motion.

3 questions · Challenging +1.3

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Edexcel M3 2005 January Q7
14 marks Challenging +1.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 6} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-6_451_1360_296_356}
\end{figure} A trapeze artiste of mass 60 kg is attached to the end \(A\) of a light inextensible rope \(O A\) of length 5 m . The artiste must swing in an arc of a vertical circle, centre \(O\), from a platform \(P\) to another platform \(Q\), where \(P Q\) is horizontal. The other end of the rope is attached to the fixed point \(O\) which lies in the vertical plane containing \(P Q\), with \(\angle P O Q = 120 ^ { \circ }\) and \(O P = O Q = 5 \mathrm {~m}\), as shown in Figure 6. As part of her act, the artiste projects herself from \(P\) with speed \(\sqrt { } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the rope \(O A\) and in the plane \(P O Q\). She moves in a circular arc towards \(Q\). At the lowest point of her path she catches a ball of mass \(m \mathrm {~kg}\) which is travelling towards her with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and parallel to \(Q P\). After catching the ball, she comes to rest at the point \(Q\). By modelling the artiste and the ball as particles and ignoring her air resistance, find
  1. the speed of the artiste immediately before she catches the ball,
  2. the value of \(m\),
  3. the tension in the rope immediately after she catches the ball.
Edexcel M3 2008 June Q5
15 marks Standard +0.8
  1. 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 released from rest with the string taut and \(O P\) horizontal.
    1. Find the tension in the string when \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical.
    A particle \(Q\) of mass \(3 m\) is at rest at a distance \(a\) vertically below \(O\). When \(P\) strikes \(Q\) the particles join together and the combined particle of mass \(4 m\) starts to move in a vertical circle with initial speed \(u\).
  2. Show that \(u = \sqrt { } \left( \frac { g a } { 8 } \right)\). The combined particle comes to instantaneous rest at \(A\).
  3. Find
    1. the angle that the string makes with the downward vertical when the combined particle is at \(A\),
    2. the tension in the string when the combined particle is at \(A\).
      \section*{LU
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OCR MEI M3 2016 June Q4
18 marks Challenging +1.2
4 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 . Particle P is projected so that it moves in complete vertical circles with centre O ; there is no air resistance. A and B are two points on the circle, situated on opposite sides of the vertical through O . The lines OA and OB make angles \(\alpha\) and \(\beta\) with the upward vertical as shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-5_414_399_434_833} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The speed of P at A is \(\sqrt { \frac { 17 a g } { 3 } }\). The speed of P at B is \(\sqrt { 5 a g }\) and \(\cos \beta = \frac { 2 } { 3 }\).
  1. Show that \(\cos \alpha = \frac { 1 } { 3 }\). On one occasion, when P is at its lowest point and moving in a clockwise direction, it collides with a stationary particle Q . The two particles coalesce and the combined particle continues to move in the same vertical circle. When this combined particle reaches the point A , the string becomes slack.
  2. Show that when the string becomes slack, the speed of the combined particle is \(\sqrt { \frac { a g } { 3 } }\). The mass of the particle Q is \(k m\).
  3. Find the value of \(k\).
  4. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.