3.03k Connected particles: pulleys and equilibrium

A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8 \text{ m s}^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is 700 N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is 2100 N, find the resistance to motion of the car. [3]

\includegraphics{figure_3} A block \(B\) of mass 0.4 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).

1. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table. [5]
2. A horizontal force of magnitude \(X\) N, acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\). [3]

\includegraphics{figure_3} The diagram shows a small block \(B\), of mass \(3\) kg, and a particle \(P\), of mass \(0.8\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. \(B\) is held at rest on a horizontal surface, and \(P\) lies on a smooth plane inclined at \(30°\) to the horizontal. When \(B\) is released from rest it accelerates at \(0.2\) m s\(^{-2}\) towards the pulley.

1. By considering the motion of \(P\), show that the tension in the string is \(3.76\) N. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]

\includegraphics{figure_6} Two particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at \(30°\) to the horizontal. \(P\) has mass \(0.2\text{ kg}\) and is held at rest on the plane. \(Q\) has mass \(0.2\text{ kg}\) and hangs freely. The string is taut (see diagram). The coefficient of friction between \(P\) and the plane is \(0.4\). The particle \(P\) is released.

1. State the tension in the string before \(P\) is released, and find the tension in the string after \(P\) is released. [6]

\(Q\) strikes the floor and remains at rest. \(P\) continues to move up the plane for a further distance of \(0.8\text{ m}\) before it comes to rest. \(P\) does not reach the pulley.

1. Find the speed of the particles immediately before \(Q\) strikes the floor. [5]
2. Calculate the magnitude of the contact force exerted on \(P\) by the plane while \(P\) is in motion. [3]

\includegraphics{figure_3} Fig. 3 shows a block of mass 15 kg on a rough, horizontal plane. A light string is fixed to the block at A, passes over a smooth, fixed pulley B and is attached at C to a sphere. The section of the string between the block and the pulley is inclined at 40° to the horizontal and the section between the pulley and the sphere is vertical. The system is in equilibrium and the tension in the string is 58.8 N.

1. The sphere has a mass of \(m\) kg. Calculate the value of \(m\). [2]
2. Calculate the frictional force acting on the block. [3]
3. Calculate the normal reaction of the plane on the block. [3]

A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1. \includegraphics{figure_6_1}

1. When the man is descending with an acceleration 1.5 m s\(^{-2}\) downwards, how much time does it take for his speed to increase from 0.5 m s\(^{-1}\) downwards to 3.5 m s\(^{-1}\) downwards? How far does he descend in this time? [4]

The man has a mass of 80 kg. All resistances to motion may be neglected.

1. Calculate the tension in the wire when the man is being lowered
   1. with an acceleration of 1.5 m s\(^{-2}\) downwards,
   2. with an acceleration of 1.5 m s\(^{-2}\) upwards. [5]

Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.

1. For safety reasons, the tension in the wire should not exceed 2500 N. What is the maximum acceleration allowed when the man is being raised? [4]

At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N. \includegraphics{figure_6_2}

1. Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the tension in the wire. [4]

\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]

A particle \(P\), of mass \(6\) kg, is attached to one end of a light inextensible string. The string passes through a small smooth ring, fixed at a point \(O\). A second particle \(Q\), of mass \(8\) kg, is attached to the other end of the string. The particle \(Q\) hangs at rest vertically below the ring, and the particle \(P\) moves with speed \(5 \text{ m s}^{-1}\) in a horizontal circle, as shown in the diagram. The angle between \(OP\) and the vertical is \(\theta\). \includegraphics{figure_4}

1. Find the tension in the string. [1 mark]
2. Find \(\theta\). [3 marks]
3. Find the radius of the horizontal circle. [4 marks]

A car, of mass 1100 kg, pulls a trailer of mass 550 kg along a straight horizontal road by means of a rigid tow-bar. The car is accelerating at 1.2 ms\(^{-2}\) and the resistances to the motion of the car and trailer have magnitudes 500 N and 200 N respectively.

1. Show that the driving force produced by the engine of the car is 2680 N. [3 marks]
2. Find the tension in the tow-bar between the car and the trailer. [3 marks]
3. Find the rate, in kW, at which the car's engine is working when the car is moving with speed 18 ms\(^{-1}\). [2 marks]

When the car is moving at 18 ms\(^{-1}\) it starts to climb a straight hill which is inclined at \(6°\) to the horizontal. If the car's engine continues to work at the same rate and the resistances to motion remain the same as previously,

1. find the acceleration of the car at the instant when it starts to climb the hill. [3 marks]
2. Show that tension in the tow-bar remains unchanged. [3 marks]

The radius of the Earth is \(R\) m. The force of attraction towards the centre of the Earth experienced by a body of mass \(m\) kg at a distance \(x\) m from the centre is of magnitude \(\frac{km}{x^2}\) N, where \(k\) is a constant.

1. Show that \(k = gR^2\). [1 mark]

Two satellites \(A\) and \(B\), each of mass \(m\) kg, are moving in circular orbits around the Earth at distances \(3R\) m and \(4R\) m respectively from the centre of the Earth. Given that the satellites move in the same plane and that they lie along the same radial line from the centre at any time,

1. show that the ratio of the speed of \(B\) to that of \(A\) is \(4:3\). [2 marks]

If, in addition, the satellites are linked with a taut, straight wire in the same plane and along the same radial line,

1. find, in terms of \(m\) and \(g\), the magnitude of the force in the wire. [8 marks]

The diagram shows two identical particles, each of mass \(m\) kg, connected by a thin, light inextensible string. \(P\) slides on the surface of a smooth right circular cylinder fixed with its axis, through \(O\), horizontal. \(Q\) moves vertically. \(OP\) makes an angle \(\theta\) radians with the horizontal. \includegraphics{figure_6} The system is released from rest in the position where \(\theta = 0\).

1. Show that the vertical distance moved by \(Q\) is \(\frac{\theta}{\sin \theta}\) times the vertical distance moved by \(P\). [4 marks]
2. In the position where \(\theta = \frac{\pi}{6}\), prove that the reaction of the cylinder on \(P\) has magnitude \(\left(1-\frac{\pi}{6}\right)mg\) N. [9 marks]

A uniform circular pulley, of mass \(4m\) and radius \(r\), is free to rotate about a fixed smooth horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. A light inextensible string passes over the pulley and has a particle of mass \(2m\) attached to one end and a particle of mass \(3m\) attached to the other end. The particles hang with the string vertical and taut on each side of the pulley. The rim of the pulley is sufficiently rough to prevent the string slipping. The system is released from rest.

1. Find the angular acceleration of the pulley. [8]

When the angular speed of the pulley is \(\Omega\), the string breaks and a constant braking couple of magnitude \(G\) is applied to the pulley which brings it to rest.

1. Find an expression for the angle turned through by the pulley from the instant when the string breaks to the instant when the pulley first comes to rest. [4]

A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after \(2.4\) seconds. The horizontal component of the initial velocity of \(P\) is \(\frac{5}{3}d \text{ m s}^{-1}\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground. [1]
2. Find the vertical component of the initial velocity of \(P\). [2]

\(P\) just clears a vertical wall which is situated at a horizontal distance \(d\) m from \(O\).

1. Find the height of the wall. [3]

The speed of \(P\) as it passes over the wall is \(16 \text{ m s}^{-1}\).

1. Find the value of \(d\) correct to \(3\) significant figures. [4]

\includegraphics{figure_9} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg. The other end of the string is attached to a second particle \(B\) of mass 2.5 kg. Particle \(A\) is in contact with a rough plane inclined at \(\theta\) to the horizontal, where \(\cos \theta = \frac{4}{5}\). The string is taut and passes over a small smooth pulley \(P\) 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. Particle \(B\) hangs freely below \(P\) at a distance 1.5 m above horizontal ground, as shown in the diagram. The coefficient of friction between \(A\) and the plane is \(\mu\). The system is released from rest and in the subsequent motion \(B\) hits the ground before \(A\) reaches \(P\). The speed of \(B\) at the instant that it hits the ground is \(1.2\) ms\(^{-1}\).

1. For the motion before \(B\) hits the ground, show that the acceleration of \(B\) is \(0.48\) ms\(^{-2}\). [1]
2. For the motion before \(B\) hits the ground, show that the tension in the string is \(23.3\) N. [3]
3. Determine the value of \(\mu\). [5] After \(B\) hits the ground, \(A\) continues to travel up the plane before coming to instantaneous rest before it reaches \(P\).
4. Determine the distance that \(A\) travels from the instant that \(B\) hits the ground until \(A\) comes to instantaneous rest. [4]

\includegraphics{figure_14} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg. The other end of the string is attached to a second particle \(B\) of mass 3 kg. Particle \(A\) is in contact with a smooth plane inclined at 30° to the horizontal and particle \(B\) is in contact with a rough horizontal plane. A second light inextensible string is attached to \(B\). The other end of this second string is attached to a third particle \(C\) of mass 4 kg. Particle \(C\) is in contact with a smooth plane \(\Pi\) inclined at an angle of 60° to the horizontal. Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from \(A\) to the pulley, and from \(C\) to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram). The coefficient of friction between \(B\) and the horizontal plane is \(\mu\). The system is released from rest and in the subsequent motion \(C\) moves down \(\Pi\) with acceleration \(a\) m s\(^{-2}\).

1. By considering an equation involving \(\mu\), \(a\) and \(g\) show that \(a < \frac{1}{9}g(2\sqrt{3} - 1)\). [7]
2. Given that \(a = \frac{1}{5}g\), determine the magnitude of the contact force between \(B\) and the horizontal plane. Give your answer correct to 3 significant figures. [4]

\includegraphics{figure_13} The diagram shows a small block \(B\), of mass \(2 \text{kg}\), and a particle \(P\), of mass \(4 \text{kg}\), which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60°\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.

1. Determine, in terms of \(g\), the tension in the string. [7]

When \(P\) is moving at \(2 \text{ms}^{-1}\) the string breaks. In the \(0.5\) seconds after the string breaks \(P\) moves \(1.9 \text{m}\) down the plane.

1. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures. [5]

In this question use \(g = 9.81\,\mathrm{m}\,\mathrm{s}^{-2}\) Two particles, of mass \(1.8\,\mathrm{kg}\) and \(1.2\,\mathrm{kg}\), are connected by a light, inextensible string over a smooth peg. \includegraphics{figure_14}

1. Initially the particles are held at rest \(1.5\,\mathrm{m}\) above horizontal ground and the string between them is taut. The particles are released from rest. Find the time taken for the \(1.8\,\mathrm{kg}\) particle to reach the ground. [5 marks]
2. State one assumption you have made in answering part (a). [1 mark]

A tractor and its driver have a combined mass of \(m\) kilograms. The tractor is towing a trailer of mass \(4m\) kilograms in a straight line along a horizontal road. The tractor and trailer are connected by a horizontal tow bar, modelled as a light rigid rod. A driving force of \(11080 \text{N}\) and a total resistance force of \(160 \text{N}\) act on the tractor. A total resistance force of \(600 \text{N}\) acts on the trailer. The tractor and the trailer have an acceleration of \(0.8 \text{m s}^{-2}\)

1. Find \(m\). [3 marks]
2. Find the tension in the tow bar. [2 marks]
3. At the instant the speed of the tractor reaches \(18 \text{km h}^{-1}\) the tow bar breaks. The total resistance force acting on the trailer remains constant. Starting from the instant the tow bar breaks, calculate the time taken until the speed of the trailer reduces to \(9 \text{km h}^{-1}\) [4 marks]

A simple lifting mechanism comprises a light inextensible wire which is passed over a smooth fixed pulley. One end of the wire is attached to a rigid triangular container of mass 2 kg, which rests on horizontal ground. A load of \(m\) kg is placed in the container. The other end of the wire is attached to a particle of mass 5 kg, which hangs vertically downwards. The mechanism is initially held at rest as shown in the diagram below. \includegraphics{figure_16} The mechanism is released from rest, and the container begins to move upwards with acceleration \(a\text{ m s}^{-2}\) The wire remains taut throughout the motion.

1. Show that $$a = \left(\frac{3 - m}{m + 7}\right)g$$ [4 marks]
2. State the range of possible values of \(m\). [1 mark]
3. In this question use \(g = 9.8\text{ m s}^{-2}\) The load reaches a height of 2 metres above the ground 1 second after it is released. Find the mass of the load. [4 marks]
4. Ignoring air resistance, describe one assumption you have made in your model. [1 mark]

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]

\includegraphics{figure_17} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v\) m s\(^{-1}\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5t + 0.01t^2$$

1. Show that when \(t = 15\) their acceleration is 0.8 m s\(^{-2}\) [2 marks]
2. The car has a mass of 1500 kg The caravan has a mass of 850 kg When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N
   1. Find the total resistance force experienced by the caravan when \(t = 15\) [2 marks]
   2. Find the driving force being applied by the car when \(t = 15\) [3 marks]
3. State one assumption you have made about the tow bar. [1 mark]

A rescue van is towing a broken-down car by using a tow bar. The van and the car are moving with a constant acceleration of \(0.6 \text{ m s}^{-2}\) along a straight horizontal road as shown in the diagram below. \includegraphics{figure_18} The van has a total mass of 2780 kg The car has a total mass of 1620 kg The van experiences a driving force of \(D\) newtons. The van experiences a total resistance force of \(R\) newtons. The car experiences a total resistance force of \(0.6R\) newtons.

1. The tension in the tow bar, \(T\) newtons, may be modelled by $$T = kD - 18$$ where \(k\) is a constant. Find \(k\) [5 marks]
2. State one assumption that must be made in answering part (a). [1 mark]