Heavier particle hits ground, lighter continues upward - vertical strings

7 Loads \(A\) and \(B\), of masses 1.2 kg and 2.0 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical. \(A\) is released and starts to move upwards. It does not reach the pulley in the subsequent motion.

1. Find the acceleration of \(A\) and the tension in the string.
2. Find, for the first 1.5 metres of \(A\) 's motion,
   1. A's gain in potential energy,
   2. the work done on \(A\) by the tension in the string,
   3. A's gain in kinetic energy. B hits the floor 1.6 seconds after \(A\) is released. \(B\) comes to rest without rebounding and the string becomes slack.
   4. Find the time from the instant the string becomes slack until it becomes taut again.

7 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-4_506_471_255_836} Two particles \(A\) and \(B\) have masses 0.12 kg and 0.38 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest with the string taut and both straight parts of the string vertical. \(A\) and \(B\) are each at a height of 0.65 m above horizontal ground (see diagram). \(A\) is released and \(B\) moves downwards. Find

1. the acceleration of \(B\) while it is moving downwards,
2. the speed with which \(B\) reaches the ground and the time taken for it to reach the ground. \(B\) remains on the ground while \(A\) continues to move with the string slack, without reaching the pulley. The string remains slack until \(A\) is at a height of 1.3 m above the ground for a second time. At this instant \(A\) has been in motion for a total time of \(T \mathrm {~s}\).
3. Find the value of \(T\) and sketch the velocity-time graph for \(A\) for the first \(T \mathrm {~s}\) of its motion.
4. Find the total distance travelled by \(A\) in the first \(T\) s of its motion.

6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.

1. Find
   1. the tension in the string before the particle of mass 1.3 kg reaches the plane,
   2. the time taken for the particle of mass 1.3 kg to reach the plane.
   3. Find the greatest height of the particle of mass 0.7 kg above the plane.

5 Particles \(A\) and \(B\), of masses 0.9 kg and 0.6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical and with the particles at the same height above the horizontal floor. In the subsequent motion, \(B\) does not reach the pulley.

1. Find the acceleration of \(A\) and the tension in the string during the motion before \(A\) hits the floor. After \(A\) hits the floor, \(B\) continues to move vertically upwards for a further 0.3 s .
2. Find the height of the particles above the floor at the instant that they started to move.

3 Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical, and with both particles at a height of 2 m above horizontal ground. \(P\) moves vertically downwards and does not rebound when it hits the ground. At the instant that \(P\) hits the ground, \(Q\) is at the point \(X\), from where it continues to move vertically upwards without reaching the pulley. Given that \(P\) has mass 0.9 kg and that the tension in the string is 7.2 N while \(P\) is moving, find the total distance travelled by \(Q\) from the instant it first reaches \(X\) until it returns to \(X\).

5 \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-08_538_414_260_868} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley with the particles hanging freely below it. \(Q\) is held at rest with the string taut at a height of \(h \mathrm {~m}\) above a horizontal floor (see diagram). \(Q\) is now released and both particles start to move. The pulley is sufficiently high so that \(P\) does not reach it at any stage. The time taken for \(Q\) to reach the floor is 0.6 s .

1. Find the acceleration of \(Q\) before it reaches the floor and hence find the value of \(h\). \(Q\) remains at rest when it reaches the floor, and \(P\) continues to move upwards.
2. Find the velocity of \(P\) at the instant when \(Q\) reaches the floor and the total time taken from the instant at which \(Q\) is released until the string becomes taut again.

5. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(2 m\) and \(3 m\) respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(A\) and \(B\) are above a horizontal plane, as shown in Figure 2. The system is released from rest.

1. Show that the tension in the string immediately after the particles are released is \(\frac { 12 } { 5 } m g\). After descending \(1.5 \mathrm {~m} , B\) strikes the plane and is immediately brought to rest. In the subsequent motion, \(A\) does not reach the pulley.
2. Find the distance travelled by \(A\) between the instant when \(B\) strikes the plane and the instant when the string next becomes taut. Given that \(m = 0.5 \mathrm {~kg}\),
3. find the magnitude of the impulse on \(B\) due to the impact with the plane.

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 3 kg and \(m \mathrm {~kg}\) respectively ( \(m > 3\) ). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut and the hanging parts of the string vertical. The particle \(Q\) is at a height of 10.5 m above the horizontal ground, as shown in Figure 5. The system is released from rest and \(Q\) moves downwards. In the subsequent motion \(P\) does not reach the pulley. After the system is released, the tension in the string is 33.6 N .

1. Show that the magnitude of the acceleration of \(P\) is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(m\). The system is released from rest at time \(t = 0\). At time \(T _ { 1 }\) seconds after release, \(Q\) strikes the ground and does not rebound. The string goes slack and \(P\) continues to move upwards.
3. Find the value of \(T _ { 1 }\) At time \(T _ { 2 }\) seconds after release, \(P\) comes to instantaneous rest.
4. Find the value of \(T _ { 2 }\) At time \(T _ { 3 }\) seconds after release ( \(T _ { 3 } > T _ { 1 }\) ) the string becomes taut again.
5. Sketch a velocity-time graph for the motion of \(P\) in the interval \(0 \leqslant t \leqslant T _ { 3 }\)

7 \includegraphics[max width=\textwidth, alt={}, center]{db77a63a-6ff8-4fe5-bdd0-15afb7eb4866-4_419_419_274_735} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are released from rest, with the string taut, and \(A\) and \(B\) at the same height above a horizontal floor (see diagram). In the subsequent motion, \(A\) descends with acceleration \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and strikes the floor 0.8 s after being released. It is given that \(B\) never reaches the pulley.

1. Calculate the distance \(A\) moves before it reaches the floor and the speed of \(A\) immediately before it strikes the floor.
2. Show that \(B\) rises a further 0.064 m after \(A\) strikes the floor, and calculate the total length of time during which \(B\) is rising.
3. Sketch the ( \(t , v\) ) graph for the motion of \(B\) from the instant it is released from rest until it reaches a position of instantaneous rest.
4. Before \(A\) strikes the floor the tension in the string is 5.88 N . Calculate the mass of \(A\) and the mass of \(B\).
5. The pulley has mass 0.5 kg , and is held in a fixed position by a light vertical chain. Calculate the tension in the chain
   1. immediately before \(A\) strikes the floor,
   2. immediately after \(A\) strikes the floor.

3. \begin{figure}[h] \end{figure} A ball \(P\) of mass \(2 m\) is attached to one end of a string.
The other end of the string is attached to a ball \(Q\) of mass \(5 m\).
The string passes over a fixed pulley.
The system is held at rest with the balls hanging freely and the string taut.
The hanging parts of the string are vertical with \(P\) at a height \(2 h\) above horizontal ground and with \(Q\) at a height \(h\) above the ground, as shown in Figure 1. The system is released from rest.
In the subsequent motion, \(Q\) does not rebound when it hits the ground and \(P\) does not hit the pulley. The balls are modelled as particles.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Air resistance is modelled as being negligible.
Using this model,

1. 1. write down an equation of motion for \(P\),
   2. write down an equation of motion for \(Q\),
2. find, in terms of \(h\) only, the height above the ground at which \(P\) first comes to instantaneous rest.
3. State one limitation of modelling the balls as particles that could affect your answer to part (b). In reality, the string will not be inextensible.
4. State how this would affect the accelerations of the particles.

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10 Particles \(P\) and \(Q\), of masses 3 kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(P\) and \(Q\) are above a horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-6_428_208_932_932}

1. Find the tension in the string immediately after the particles are released. After descending \(2.5 \mathrm {~m} , Q\) strikes the plane and is immediately brought to rest. It is given that \(P\) does not reach the pulley in the subsequent motion.
2. Find the distance travelled by \(P\) between the instant when \(Q\) strikes the plane and the instant when the string becomes taut again.

5. \begin{figure}[h] \end{figure} Two particles, \(P\) and \(Q\), have masses 3 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a small smooth fixed pulley. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 3. Immediately after the particles are released from rest, \(P\) moves upwards with acceleration \(a \mathrm {~ms} ^ { - 2 }\) and the tension in the string is \(T\) newtons.

1. Write down an equation of motion for \(P\).
2. Find the value of \(T\). The total force acting on the pulley due to the string has magnitude \(F\) newtons.
3. Find the value of \(F\). Initially, \(Q\) is 10 m above horizontal ground and \(P\) is more than 2 m below the pulley.
   At the instant when \(Q\) has descended a distance of 2 m , the string breaks and \(Q\) falls to the ground.
4. Find the speed of \(Q\) at the instant it hits the ground.

Two particles \(A\) and \(B\), of masses \(2.4\text{kg}\) and \(1.2\text{kg}\) respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at a distance of \(2.1\text{m}\) above a horizontal plane and \(B\) is \(1.5\text{m}\) above the plane. The particles hang vertically and are released from rest. In the subsequent motion \(A\) reaches the plane and does not rebound and \(B\) does not reach the pulley.

1. Show that the tension in the string before \(A\) reaches the plane is \(16\text{N}\) and find the magnitude of the acceleration of the particles before \(A\) reaches the plane. [4]
2. Find the greatest height of \(B\) above the plane. [3]

\includegraphics{figure_5} Two particles of masses 0.8 kg and 0.2 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The system is released from rest with both particles 0.5 m above a horizontal floor (see diagram). In the subsequent motion the 0.2 kg particle does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(6 \text{ m s}^{-2}\) and find the tension in the string. [4]
2. When the 0.8 kg particle reaches the floor it comes to rest. Find the greatest height of the 0.2 kg particle above the floor. [3]

\includegraphics{figure_7} Two particles, \(A\) and \(B\), of masses 3 kg and 5 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held with the string taut and its straight parts vertical. Particle \(A\) is 1 m above a horizontal plane, and particle \(B\) is 2 m above the plane (see diagram). The particles are released from rest. In the subsequent motion, \(A\) does not reach the pulley, and after \(B\) reaches the plane it remains in contact with the plane.

1. Find the tension in the string and the time taken for \(B\) to reach the plane. [6]
2. Find the time for which \(A\) is at least 3.25 m above the plane. [4]

\includegraphics{figure_6} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is held at rest with the string taut and its straight parts vertical. Both particles are at a height of 0.36 m above the floor (see diagram). The system is released and \(A\) begins to fall, reaching the floor after 0.6 s.

1. Find the acceleration of \(A\) as it falls. [2]

The mass of \(A\) is 0.45 kg. Find

1. the tension in the string while \(A\) is falling, [2]
2. the mass of \(B\), [3]
3. the maximum height above the floor reached by \(B\). [3]

\includegraphics{figure_5} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string which passes over a fixed smooth pulley. Both \(A\) and \(B\) are 0.5 m above the ground. The particles hang vertically (see diagram). The particles are released from rest. In the subsequent motion \(B\) does not reach the pulley and \(A\) remains at rest after reaching the ground.

1. For the motion before \(A\) reaches the ground, show that the magnitude of the acceleration of each particle is \(\frac{10}{3}\) m s\(^{-2}\) and find the tension in the string. [4]
2. Find the maximum height of \(B\) above the ground. [4]

\includegraphics{figure_6} Two particles of masses \(1.2\) kg and \(0.8\) kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest with both particles \(0.64\) m above the floor (see diagram). In the subsequent motion the \(0.8\) kg particle does not reach the pulley.

1. Show that the acceleration of the particles is \(2\) m s\(^{-2}\) and find the tension in the string. [4]
2. Find the total distance travelled by the \(0.8\) kg particle during the first second after the particles are released. [8]

\includegraphics{figure_4} Two particles \(A\) and \(B\) have masses \(5m\) and \(km\) respectively, where \(k < 5\). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut, the hanging parts of the string vertical and with \(A\) and \(B\) at the same height above a horizontal plane, as shown in Figure 4. The system is released from rest. After release, \(A\) descends with acceleration \(\frac{1}{4}g\).

1. Show that the tension in the string as \(A\) descends is \(\frac{15}{4}mg\). [3]
2. Find the value of \(k\). [3]
3. State how you have used the information that the pulley is smooth. [1]

After descending for 1.2 s, the particle \(A\) reaches the plane. It is immediately brought to rest by the impact with the plane. The initial distance between \(B\) and the pulley is such that, in the subsequent motion, \(B\) does not reach the pulley.

1. Find the greatest height reached by \(B\) above the plane. [7]

\includegraphics{figure_2} Figure 2 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, attached to the ends of a light, inextensible string which passes over a smooth, fixed pulley. The system is released from rest with \(P\) and \(Q\) at the same level 1.5 metres above the ground and 2 metres below the pulley.

1. Show that the initial acceleration of the system is \(\frac{g}{5}\) m s\(^{-2}\). [4 marks]
2. Find the tension in the string. [2 marks]
3. Find the speed with which \(P\) hits the ground. [3 marks]

When \(P\) hits the ground, it does not rebound.

1. What is the closest that \(Q\) gets to the pulley. [5 marks]

A light inextensible string passes over a fixed smooth light pulley. Particles \(A\) and \(B\), of masses 2 kg and 3 kg respectively, are attached to the ends so that the portions of the string below the axis of the pulley are vertical (see diagram). The centre of the horizontal axis of the pulley is 4 m above ground level. \includegraphics{figure_13} The particles are released from rest 1 m above ground level with the string taut.

1. Determine the acceleration of both particles prior to the impact of \(B\) with the ground. [3]
2. Determine the greatest height attained by \(A\) above ground level. [3]
3. If \(B\) rebounds after impact to a first maximum height of 0.05 m, determine the coefficient of restitution between \(B\) and the ground. [2]

A light inextensible string passes over a smooth fixed pulley. Particles of mass 0.2 kg and 0.3 kg are attached to opposite ends of the string, so that the parts of the string not in contact with the pulley are vertical. The system is released from rest with the string taut.

1. Find the acceleration of the particles and the tension in the string. [6]

When the heavier particle has fallen 2.25 m it hits the ground and is brought to rest (and the string goes slack).

1. Find the speed with which it hits the ground. [2]
2. Find the magnitude of the impulse of the ground on the particle. [2]
3. If the impact between the particle and the ground lasts for 0.005 seconds, find the constant force that would be needed to bring the particle to rest. [2]