Lighter particle on surface released, heavier hangs

6 \includegraphics[max width=\textwidth, alt={}, center]{efa7175f-832b-4cd3-82ab-52e402115081-4_686_511_269_817} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is held on the horizontal floor and particle \(B\) hangs in equilibrium. Particle \(A\) is released and both particles start to move vertically.

1. Find the acceleration of the particles. The speed of the particles immediately before \(B\) hits the floor is \(1.6 \mathrm {~ms} ^ { - 1 }\). Given that \(B\) does not rebound upwards, find
2. the maximum height above the floor reached by \(A\),
3. the time taken by \(A\), from leaving the floor, to reach this maximum height.

2 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_569_519_676_813} Particle \(A\) of mass 0.2 kg and particle \(B\) of mass 0.6 kg are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(B\) is held at rest at a height of 1.6 m above the floor. \(A\) hangs freely at a height of \(h \mathrm {~m}\) above the floor. Both straight parts of the string are vertical (see diagram). \(B\) is released and both particles start to move. When \(B\) reaches the floor it remains at rest, but \(A\) continues to move vertically upwards until it reaches a height of 3 m above the floor. Find the speed of \(B\) immediately before it hits the floor, and hence find the value of \(h\).

4 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-3_442_495_255_826} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is at rest with \(B\) held on the horizontal floor, the string taut and its straight parts vertical. \(A\) is at a height of 1.6 m above the floor (see diagram). \(B\) is released and the system begins to move; \(B\) does not reach the pulley. Find

1. the acceleration of the particles and the tension in the string before \(A\) reaches the floor,
2. the greatest height above the floor reached by \(B\).

4 Two particles \(A\) and \(B\) have masses 0.35 kg and 0.45 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small fixed smooth pulley which is 1 m above horizontal ground. Initially particle \(A\) is held at rest on the ground vertically below the pulley, with the string taut. Particle \(B\) hangs vertically below the pulley at a height of 0.64 m above the ground. Particle \(A\) is released.

1. Find the speed of \(A\) at the instant that \(B\) reaches the ground.
2. Assuming that \(B\) does not bounce after it reaches the ground, find the total distance travelled by \(A\) between the instant that \(B\) reaches the ground and the instant when the string becomes taut again.

5 Two particles are connected by a light inextensible string that passes over a smooth peg. The particles have masses of 3 kg and 1 kg . The 1 kg particle is pulled down to ground level, where it is 40 cm below the level of the 3 kg particle, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-3_490_648_1272_696} The particles are released from rest with the string vertical above each particle. Assume that no resistance forces act on the particles as they move.

1. By forming two equations of motion, one for each particle, find the magnitude of the acceleration of the particles after they have been released but before the 3 kg particle hits the ground.
2. Find the speed of the 1 kg particle when the 3 kg particle hits the ground.
3. After the 3 kg particle has hit the ground, the 1 kg particle continues to move and the string is now slack. Find the maximum height above ground level reached by the 1 kg particle.
4. If a constant air resistance force also acts on the particles as they move, explain how this would change your answer for the acceleration in part (a). Give a reason for your answer.

5 Two particles, of masses 3 kg and 7 kg , are connected by a light inextensible string that passes over a smooth peg. The 3 kg particle is held at ground level with the string above it taut and vertical. The 7 kg particle is at a height of 80 cm above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-10_469_600_486_721} The 3 kg particle is then released from rest.

1. By forming two equations of motion, show that the magnitude of the acceleration of the particles is \(3.92 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the speed of the 7 kg particle just before it hits the ground.
3. When the 7 kg particle hits the ground, the string becomes slack and in the subsequent motion the 3 kg particle does not hit the peg. Find the maximum height of the 3 kg particle above the ground.
   [0pt] [4 marks]

7.
A particle \(P\), of mass 4 kg , rests on horizontal ground and is attached by a light, inextensible string to another particle \(Q\) of mass 4.5 kg . The string passes over a smooth pulley whose centre is 3 m above the ground. Initially \(Q\) is 1.1 m below the level of the centre of the pulley. The system is released from rest in this position.

1. Find the acceleration of the two particles.
2. Find the speed with which \(Q\) hits the ground. Assuming that \(Q\) does not rebound from the ground while the string is slack,
3. show that \(P\) does not reach the pulley before \(Q\) starts to move again.
4. Find the speed with which \(Q\) leaves the ground when the string again becomes taut.
   (3 marks)