3.03o Advanced connected particles: and pulleys

\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_5} Figure 5 shows two particles \(A\) and \(B\), of mass \(2m\) and \(4m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,

1. give a reason why the magnitudes of the accelerations of the two particles are the same, [1]
2. write down an equation of motion for each particle, [4]
3. find the acceleration of each particle. [5]

Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),

1. find the distance \(XY\) in terms of \(h\). [6]

\includegraphics{figure_3} Particles \(A\) and \(B\), of mass \(2m\) and \(m\) respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle \(A\) is held on the table, while \(B\) rests on a smooth plane inclined at \(30°\) to the horizontal, as shown in Fig. 3. The string is in the same vertical plane as a line of greatest slope of the inclined plane. The coefficient of friction between \(A\) and the table is \(\mu\). The particle \(A\) is released from rest and begins to move. By writing down an equation of motion for each particle,

1. show that, while both particles move with the string taut. Each particle has an acceleration of magnitude \(\frac{1}{5}(1 - 4\mu)g\). [7]

When each particle has moved a distance \(h\), the string breaks. The particle \(A\) comes to rest before reaching the pulley. Given that \(\mu = 0.2\),

1. find, in terms of \(h\), the total distance moved by \(A\). [6]

For the model described above,

1. state two physical factors, apart from air resistance, which could be taken into account to make the model more realistic. [2]

Two metal weights \(A\) and \(B\), of masses 2.4 kg and 1.8 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley so that the string hangs vertically on each side. The system is released from rest with the string taut.

1. Calculate the acceleration of each weight and the tension in the string. [6 marks]

\(A\) is now replaced by a different weight of mass \(m\) kg, where \(m < 1.8\), and the system is again released from rest. The magnitude of the acceleration has half of its previous value.

1. Calculate the value of \(m\). [6 marks]

A particle \(P\) of mass \(m\) kg, at rest on a smooth horizontal table, is connected to particles \(Q\) and \(R\), of mass 0.1 kg and 0.5 kg respectively, by strings which pass over fixed pulleys at the edges of the table. The system is released from rest with \(Q\) and \(R\) hanging freely and it is found that the tension in the section of the string between \(P\) and \(R\) is 2 N.

1. Show that the acceleration of the particles has magnitude 5.8 ms\(^{-2}\). \hfill [3 marks]
2. Find the value of \(m\). \hfill [5 marks]

Modelling assumptions have been made about the pulley and the strings.

1. Briefly describe these two assumptions. For each one, state how the mathematical model would be altered if the assumption were not made. \hfill [4 marks]

\includegraphics{figure_1} Particles \(P\) and \(Q\), of masses \(0.3\) kg and \(0.4\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in motion with the string taut and with each of the particles moving vertically. The downward acceleration of \(P\) is \(a\) m s\(^{-2}\) (see diagram).

1. Show that \(a = -1.4\). [4]

Initially \(P\) and \(Q\) are at the same horizontal level. \(P\)'s initial velocity is vertically downwards and has magnitude \(2.8\) m s\(^{-1}\).

1. Assuming that \(P\) does not reach the floor and that \(Q\) does not reach the pulley, find the time taken for \(P\) to return to its initial position. [3]

\includegraphics{figure_6} A train of total mass \(80000\) kg consists of an engine \(E\) and two trucks \(A\) and \(B\). The engine \(E\) and truck \(A\) are connected by a rigid coupling \(X\), and trucks \(A\) and \(B\) are connected by another rigid coupling \(Y\). The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on \(E\), \(A\) and \(B\) are \(10500\) N, \(3000\) N and \(1500\) N respectively (see diagram).

1. By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than \(15000\) N. [2]
2. Show that, when the magnitude of the driving force is \(35000\) N, the acceleration of the train is \(0.25\) m s\(^{-2}\). [2]
3. Hence find the mass of \(E\), given that the tension in the coupling \(X\) is \(8500\) N when the magnitude of the driving force is \(35000\) N. [3]

The driving force is replaced by a braking force of magnitude \(15000\) N acting on the engine. The force exerted by the coupling \(Y\) is zero.

1. Find the mass of \(B\). [5]
2. Show that the coupling \(X\) exerts a forward force of magnitude \(1500\) N on the engine. [2]

\includegraphics{figure_3} Figure 3 shows a particle \(X\) of mass 3 kg on a smooth plane inclined at an angle 30° to the horizontal, and a particle \(Y\) of mass 2 kg on a smooth plane inclined at an angle 60° to the horizontal. The two particles are connected by a light, inextensible string of length 2.5 metres passing over a smooth pulley at \(C\) which is the highest point of the two planes. Initially, \(Y\) is at a point just below \(C\) touching the pulley with the string taut. When the particles are released from rest they travel along the lines of greatest slope, \(AC\) in the case of \(X\) and \(BC\) in the case of \(Y\), of their respective planes. \(A\) and \(B\) are the points where the planes meet the horizontal ground and \(AB = 4\) metres.

1. Show that the initial acceleration of the system is given by \(\frac{g}{10}\left(2\sqrt{3} - 3\right)\) ms\(^{-2}\). [7 marks]
2. By finding the tension in the string, or otherwise, find the magnitude of the force exerted on the pulley and the angle that this force makes with the vertical. [7 marks]
3. Find, correct to 2 decimal places, the speed with which \(Y\) hits the ground. [4 marks]

\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]

\includegraphics{figure_2} Figure 2 shows a particle \(A\) of mass 5 kg, lying on a smooth horizontal table which is 0.9 m above the floor. A light inextensible string of length 0.7 m connects \(A\) to a particle \(B\) of mass 2 kg. The string passes over a smooth pulley which is fixed to the edge of the table and \(B\) hangs vertically 0.4 m below the pulley. When the system is released from rest,

1. show that the magnitude of the force exerted on the pulley is \(\frac{10\sqrt{5}}{7}\) g N. [7 marks]
2. find the speed with which \(A\) hits the pulley. [3 marks]

When \(A\) hits the pulley, the string breaks and \(B\) subsequently falls freely under gravity.

1. Find the speed with which \(B\) hits the ground. [4 marks]

\includegraphics{figure_3} Figure 3 shows two particles \(A\) and \(B\) of masses \(m\) and \(km\) respectively, connected by a light inextensible string which passes over a smooth fixed pulley. When the system is released from rest with both particles 0.5 m above the ground, particle \(A\) moves vertically upwards with acceleration \(\frac{1}{4} g \text{ m s}^{-2}\).

1. Write down, with a brief justification, the magnitude and direction of the acceleration of \(B\). [2 marks]
2. Find the value of \(k\). [6 marks] Given that \(A\) does not hit the pulley,
3. calculate, correct to 3 significant figures, the speed with which \(B\) hits the ground. [3 marks]

The diagram shows two objects \(A\) and \(B\), of mass 3 kg and 5 kg respectively, connected by a light inextensible string passing over a light smooth pulley fixed at the end of a smooth horizontal surface. Object \(A\) lies on the horizontal surface and object \(B\) hangs freely below the pulley. \includegraphics{figure_8} Initially, \(B\) is supported so that the objects are at rest with the string just taut. Object \(B\) is then released.

1. Find the magnitude of the acceleration of \(A\) and the tension in the string. [6]
2. State briefly what effect a rough pulley would have on the tension in the string. [1]

The diagram below shows an object \(A\), of mass \(2m\) kg, lying on a horizontal table. It is connected to another object \(B\), of mass \(m\) kg, by a light inextensible string, which passes over a smooth pulley \(P\), fixed at the edge of the table. Initially, object \(A\) is held at rest so that object \(B\) hangs freely with the string taut. \includegraphics{figure_9} Object \(A\) is then released.

1. When object \(B\) has moved downwards a vertical distance of 0·4 m, its speed is 1·2 ms\(^{-1}\). Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of \(B\) is 1·8 ms\(^{-2}\). [2]
2. During the motion, object \(A\) experiences a constant resistive force of 22 N. Find the value of \(m\) and hence determine the tension in the string. [6]
3. What assumption did the word 'inextensible' in the description of the string enable you to make in your solution? [1]

Particle \(A\), of mass \(m\) kg, lies on the plane \(\Pi_1\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. Particle \(B\), of \(4m\) kg, lies on the plane \(\Pi_2\) inclined at an angle of \(\tan^{-1} \frac{4}{3}\) to the horizontal. The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\). The coefficient of friction between particle \(A\) and \(\Pi_1\) is \(\frac{1}{4}\) and plane \(\Pi_2\) is smooth. Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below. \includegraphics{figure_13}

1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac{7g}{15}\) m s\(^{-2}\). [6]
2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac{1}{4}\) m when its speed is \(\sqrt{\frac{7g}{30}}\) m s\(^{-1}\). [2]

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]

\includegraphics{figure_11} The diagram shows a particle, \(A\), of mass \(m_1\) at rest on a rough slope at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). Particle \(A\) is connected by a light inextensible string to another particle, \(B\), of mass \(m_2\). The string passes over a smooth peg at the top of the slope and particle \(B\) is hanging freely.

1. In the case when \(m_2 = \frac{1}{4}m_1\), particle \(A\) is on the point of sliding down the slope.
   1. Draw a fully labelled diagram to show all the forces acting on the particles. [2]
   2. Find the coefficient of friction between \(A\) and the slope. [6]
2. In the case when \(m_2 = m_1\), find the acceleration of the particles. [4]