String breaks during motion

7 \includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-3_376_1052_1171_548} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a rough horizontal table with the string passing over a smooth pulley fixed at the edge of the table. The coefficient of friction between \(A\) and the table is 0.2 . Particle \(B\) hangs vertically below the pulley at a height of 0.5 m above the floor (see diagram). The system is released from rest and 0.25 s later the string breaks. A does not reach the pulley in the subsequent motion. Find

1. the speed of \(B\) immediately before it hits the floor,
2. the total distance travelled by \(A\).

6 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-4_712_529_264_810} Particles \(P\) and \(Q\), of masses 0.55 kg and 0.45 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The particles are held at rest with the string taut and its straight parts vertical. Both particles are at a height of 5 m above the ground (see diagram). The system is released.

1. Find the acceleration with which \(P\) starts to move. The string breaks after 2 s and in the subsequent motion \(P\) and \(Q\) move vertically under gravity.
2. At the instant that the string breaks, find
   1. the height above the ground of \(P\) and of \(Q\),
   2. the speed of the particles.
   3. Show that \(Q\) reaches the ground 0.8 s later than \(P\). \(7 \quad\) A particle \(P\) starts from rest at the point \(A\) at time \(t = 0\), where \(t\) is in seconds, and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . For \(10 \leqslant t \leqslant 20 , P\) continues to move along the line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 800 } { t ^ { 2 } } - 2\). Find
      1. the speed of \(P\) when \(t = 10\), and the value of \(a\),
      2. the value of \(t\) for which the acceleration of \(P\) is \(- a \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
      3. the displacement of \(P\) from \(A\) when \(t = 20\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-3_518_515_1436_815} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 0.52 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the tension in the string. When both particles are moving with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks.
2. Find the time taken, from the instant that the string breaks, for \(A\) to reach the floor. \(7 \quad\) A particle \(P\) starts from rest at a point \(O\) and moves in a straight line. \(P\) has acceleration \(0.6 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t\) seconds after leaving \(O\), until \(t = 10\).
3. Find the velocity and displacement from \(O\) of \(P\) when \(t = 10\). After \(t = 10 , P\) has acceleration \(- 0.4 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at a point \(A\).
4. Find the distance \(O A\).

5 A small block \(P\) is attached to another small block \(Q\) by a light inextensible string. The block \(P\) rests on a rough horizontal surface and the string hangs over a smooth peg so that \(Q\) hangs freely, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-4_222_426_507_810} The block \(P\) has mass 0.4 kg and the coefficient of friction between \(P\) and the surface is 0.5 . The block \(Q\) has mass 0.3 kg . The system is released from rest and \(Q\) moves vertically downwards.

1. 1. Draw a diagram to show the forces acting on \(P\).
   2. Show that the frictional force between \(P\) and the surface has magnitude 1.96 newtons.
2. By forming an equation of motion for each block, show that the magnitude of the acceleration of each block is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the speed of the blocks after 3 seconds of motion.
4. After 3 seconds of motion, the string breaks. The blocks continue to move. Comment on how the speed of each block will change in the subsequent motion. For each block, give a reason for your answer.

6 Two particles, \(A\) and \(B\), have masses 12 kg and 8 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg, as shown in the diagram. $$A ( 12 \mathrm {~kg} )$$ The particles are released from rest and move vertically. Assume that there is no air resistance.

1. By forming two equations of motion, show that the magnitude of the acceleration of each particle is \(1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. After the particles have been moving for 2 seconds, both particles are at a height of 4 metres above a horizontal surface. When the particles are in this position, the string breaks.
   1. Find the speed of particle \(A\) when the string breaks.
   2. Find the speed of particle \(A\) when it hits the surface.
   3. Find the time that it takes for particle \(B\) to reach the surface after the string breaks. Assume that particle \(B\) does not hit the peg.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-13_2484_1709_223_153}

12 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-5_429_873_264_635} The diagram shows a block \(B\) of mass 2 kg and a particle \(A\) of mass 3 kg attached to opposite ends of a light inextensible string. The block is held at rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal, and the coefficient of friction between the block and the plane is 0.4 . The string passes over a small smooth pulley \(C\) at the edge of the plane and \(A\) hangs in equilibrium 1.2 m above horizontal ground. The part of the string between \(B\) and \(C\) is parallel to a line of greatest slope of the plane. \(B\) is released and begins to move up the plane.

1. Show that the acceleration of \(A\) is \(3.13 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures, and find the tension in the string.
2. When \(A\) reaches the ground it remains there. Given that \(B\) does not reach \(C\) in the subsequent motion, find the total time that \(B\) is moving up the plane.

\includegraphics{figure_3} Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley.

1. Find the tension in the string immediately after the particles are released. [6]
2. Find the acceleration of \(A\) immediately after the particles are released. [2]

When the particles have been moving for 0.5 s, the string breaks.

1. Find the further time that elapses until \(B\) hits the floor. [9]

In this question, use \(g = 10\) m s⁻² A box, B, of mass 4 kg lies at rest on a fixed rough horizontal shelf. One end of a light string is connected to B. The string passes over a smooth peg, attached to the end of the shelf. The other end of the string is connected to particle, P, of mass 1 kg, which hangs freely below the shelf as shown in the diagram below. \includegraphics{figure_15} B is initially held at rest with the string taut. B is then released. B and P both move with constant acceleration \(a\) m s⁻² As B moves across the shelf it experiences a total resistance force of 5 N

1. State one type of force that would be included in the total resistance force. [1 mark]
2. Show that \(a = 1\) [4 marks]
3. When B has moved forward exactly 20 cm the string breaks. Find how much further B travels before coming to rest. [4 marks]
4. State one assumption you have made when finding your solutions in parts (b) or (c). [1 mark]