Multi-stage motion: changing surface conditions or external intervention

7 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-10_551_776_260_689} Two particles \(P\) and \(Q\), of masses 2 kg and 0.25 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(P\) is on an inclined plane at an angle of \(30 ^ { \circ }\) to the horizontal. Particle \(Q\) hangs below the pulley. Three points \(A , B\) and \(C\) lie on a line of greatest slope of the plane with \(A B = 0.8 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram). Particle \(P\) is released from rest at \(A\) with the string taut and slides down the plane. During the motion of \(P\) from \(A\) to \(C , Q\) does not reach the pulley. The part of the plane from \(A\) to \(B\) is rough, with coefficient of friction 0.3 between the plane and \(P\). The part of the plane from \(B\) to \(C\) is smooth.

1. 1. Find the acceleration of \(P\) between \(A\) and \(B\).
   2. Hence, find the speed of \(P\) at \(C\).
2. Find the time taken for \(P\) to travel from \(A\) to \(C\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-4_392_1192_255_424} \(A B\) and \(B C\) are lines of greatest slope on a fixed triangular prism, and \(M\) is the mid-point of \(B C . A B\) and \(B C\) are inclined at \(30 ^ { \circ }\) to the horizontal. The surface of the prism is smooth between \(A\) and \(B\), and between \(B\) and \(M\). Between \(M\) and \(C\) the surface of the prism is rough. A small smooth pulley is fixed to the prism at \(B\). A light inextensible string passes over the pulley. Particle \(P\) of mass 0.3 kg is fixed to one end of the string, and is placed at \(A\). Particle \(Q\) of mass 0.4 kg is fixed to the other end of the string and is placed next to the pulley on \(B C\). The particles are released from rest with the string taut. \(P\) begins to move towards the pulley, and \(Q\) begins to move towards \(M\) (see diagram).

1. Show that the initial acceleration of the particles is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and find the tension in the string. The particle \(Q\) reaches \(M 1.8 \mathrm {~s}\) after being released from rest.
2. Find the speed of the particles when \(Q\) reaches \(M\). After \(Q\) passes through \(M\), the string remains taut and the particles decelerate uniformly. \(Q\) comes to rest between \(M\) and \(C 1.4 \mathrm {~s}\) after passing through \(M\).
3. Find the deceleration of the particles while \(Q\) is moving from \(M\) towards \(C\).
4. (a) By considering the motion of \(P\), find the tension in the string while \(Q\) is moving from \(M\) towards \(C\).
   (b) Calculate the magnitude of the frictional force which acts on \(Q\) while it is moving from \(M\) towards \(C\). \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

5 A block, of mass \(3 m\), is placed on a horizontal surface at a point \(A\). A light inextensible string is attached to the block and passes over a smooth peg. The string is horizontal between the block and the peg. A particle, of mass \(2 m\), is attached to the other end of the string. The block is released from rest with the string taut and the string between the peg and the particle vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-10_170_726_536_657} Assume that there is no air resistance acting on either the block or the particle, and that the size of the block is negligible. The horizontal surface is smooth between the points \(A\) and \(B\), but rough between the points \(B\) and \(C\). Between \(B\) and \(C\), the coefficient of friction between the block and the surface is 0.8 .

1. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(A\) and \(B\).
2. Given that the distance between the points \(A\) and \(B\) is 1.2 metres, find the speed of the block when it reaches \(B\).
3. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(B\) and \(C\).
4. Given that the distance between the points \(B\) and \(C\) is 0.9 metres, find the speed of the block when it reaches \(C\).
5. Explain why it is important to assume that the size of the block is negligible.
   [0pt] [1 mark]

Two smooth spheres \(A\) and \(B\), of masses \(2m\) and \(m\) respectively, are connected by a light inextensible string which passes over a smooth fixed pulley as shown. \(A\) is initially at rest on the rough horizontal surface of a table, the coefficient of friction between \(A\) and the table being \(\frac{2}{7}\). \(B\) hangs freely on the end of the vertical portion of the string. \includegraphics{figure_5} \(A\) is now given an impulse, directed away from the pulley, of magnitude \(5m\) Ns.

1. Show that the system starts to move with speed \(2.5 \text{ ms}^{-1}\). [1 mark]
2. State which modelling assumption ensures that the tensions in the two sections of the string can be taken to be equal. [1 mark]

Given that \(A\) comes to rest before it reaches the edge of the table and before \(B\) hits the pulley,

1. find the time taken for the system to come to rest. [7 marks]
2. Find the distance travelled by \(A\) before it first comes to rest. [4 marks]

A particle \(P\) of mass \(2\) kg rests on a long rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.2\). A light inextensible string has one end attached to \(P\) and the other end attached to a particle \(Q\) of mass \(4\) kg. The particle \(Q\) is placed on a smooth plane inclined at \(30^{\circ}\) to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time \(t = 0\), where \(t\) is measured in seconds. In the subsequent motion \(P\) does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(\frac{8}{3}\) m s\(^{-2}\). [4]

After the particles have moved a distance of \(12\) m the string is cut.

1. Find the corresponding value of \(t\) and the speed of the particles at this instant. [4]
2. Find the value of \(t\) when \(P\) comes to rest. [3]