3.03k Connected particles: pulleys and equilibrium

9. \begin{figure}[h] \end{figure} A small ball \(A\) of mass 2.5 kg is held at rest on a rough horizontal table.
The ball is attached to one end of a string.
The string passes over a pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a small ball \(B\) of mass 1.5 kg hanging freely, vertically below \(P\) and with \(B\) at a height of 1 m above the horizontal floor. The system is release from rest, with the string taut, as shown in Figure 2.
The resistance to the motion of \(A\) from the rough table is modelled as having constant magnitude 12.7 N . Ball \(B\) reaches the floor before ball \(A\) reaches 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 and the acceleration due to gravity, \(g\), is modelled as being \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence find the acceleration of \(B\).
3. Using the model, find the time it takes, from release, for \(B\) to reach the floor.
4. Suggest two improvements that could be made in the model.

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.

8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_213_1095_429_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_215_1095_427_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

An aircraft moves along a straight horizontal runway with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\). Points \(A\) and \(B\) lie on the runway. The aircraft passes \(A\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and its speed at \(B\) must be at least \(78 \mathrm {~ms} ^ { - 1 }\) if it is to take off successfully. a) Find the speed of the aircraft 8 seconds after it passes \(A\).
b) Determine the minimum value of the distance \(A B\) for the aircraft to take off successfully. The diagram below shows an object \(A\), of mass 15 kg , lying on a smooth horizontal surface. It is connected to a box \(B\) by a light inextensible string which passes over a smooth pulley \(P\), fixed at the edge of the surface, so that box \(B\) hangs freely. An object \(C\) lies on the horizontal floor of box \(B\) so that the combined mass of \(B\) and \(C\) is 10 kg . \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-09_661_862_614_598} Initially, the system is held at rest with the string just taut. A horizontal force of magnitude 150 N is then applied to \(A\) in the direction \(P A\) so that box \(B\) is raised.
a) Find the magnitude of the acceleration of \(A\) and the tension in the string.
b) Given that object \(C\) has mass 4 kg , calculate the reaction of the floor of the box on object \(C\).
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1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:

• the shop has position vector \(\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }\),
• the park is 2 km away on a bearing of \(060 ^ { \circ }\).
  a) Show that the position vector of the park relative to the house is \(( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }\).
  b) Determine the total distance walked by Sarah from her house to the park.
  c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.

\includegraphics{figure_2} Two particles \(A\) and \(B\) have masses \(m\) kg and 0.1 kg respectively, where \(m > 0.1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.9 m above horizontal ground (see diagram). The system is released from rest, and while both particles are in motion the tension in the string is 1.5 N. Particle \(B\) does not reach the pulley.

1. Find \(m\). [4]
2. Find the speed at which \(A\) reaches the ground. [2]

Two particles \(P\) and \(Q\), of masses 0.5 kg and 0.3 kg respectively, are connected by a light inextensible string. The string is taut and \(P\) is vertically above \(Q\). A force of magnitude 10 N is applied to \(P\) vertically upwards. Find the acceleration of the particles and the tension in the string connecting them. [5]

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_6} Two particles \(P\) and \(Q\), of masses \(0.2\) kg and \(0.1\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley \(B\) which is attached to two inclined planes. Particle \(P\) lies on a smooth plane \(AB\) which is inclined at \(60°\) to the horizontal. Particle \(Q\) lies on a plane \(BC\) which is inclined at an angle of \(\theta°\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).

1. It is given that \(\theta = 60\), the plane \(BC\) is rough and the coefficient of friction between \(Q\) and the plane \(BC\) is \(0.7\). The particles are released from rest. Determine whether the particles move. [4]
2. It is given instead that the plane \(BC\) is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \((\sqrt{3} - 1)\) N. Find the magnitude of the acceleration of \(P\) as it moves on the plane, and find the value of \(\theta\). [4]

\includegraphics{figure_7} Two particles \(P\) and \(Q\) of masses 2.5 kg and 0.5 kg respectively are connected by a light inextensible string that passes over a small smooth pulley fixed at the top of a plane inclined at an angle of \(30°\) to the horizontal. Particle \(P\) is on the plane and \(Q\) hangs below the pulley such that the level of \(Q\) is 2 m below the level of \(P\) (see diagram). Particle \(P\) is released from rest with the string taut and slides down the plane. The plane is rough with coefficient of friction 0.2 between the plane and \(P\).

1. Find the acceleration of \(P\). [5]
2. Use an energy method to find the speed of the particles at the instant when they are at the same vertical height. [5]

A toy railway locomotive of mass 0.8 kg is towing a truck of mass 0.4 kg on a straight horizontal track at a constant speed of \(2\,\text{m}\,\text{s}^{-1}\). There is a constant resistance force of magnitude 0.2 N on the locomotive, but no resistance force on the truck. There is a light rigid horizontal coupling connecting the locomotive and the truck.

1. State the tension in the coupling. [1]
2. Find the power produced by the locomotive's engine. [1] The power produced by the locomotive's engine is now changed to 1.2 W.
3. Find the magnitude of the tension in the coupling at the instant that the locomotive begins to accelerate. [5]

A car of mass 1800 kg is towing a trailer of mass 300 kg up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800 N acting on the car and a resistance force of \(F\) N acting on the trailer. The driving force of the car's engine is 3000 N.

1. It is given that \(F = 100\). Find the acceleration of the car and the tension in the tow-bar. [5]
2. It is given instead that the total work done against \(F\) in moving a distance of 50 m up the road is 6000 J. The speed of the car at the start of the 50 m is \(20\) m s\(^{-1}\). Use an energy method to find the speed of the car at the end of the 50 m. [5]

\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]

A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at an angle of \(\sin^{-1} 0.08\) to the horizontal. The resistances to the motion of the car and trailer are 400 N and 200 N respectively. The car and trailer are connected by a light rigid tow-bar. The car and trailer have speed \(30 \text{ m s}^{-1}\) at the bottom of the hill and \(20 \text{ m s}^{-1}\) at the top of the hill.

1. Use an energy method to find the constant driving force as the car and trailer travel up the hill. [5]
2. After reaching the top of the hill the system consisting of the car and trailer travels along a straight level road. The driving force of the car's engine is 2400 N and the resistances to motion are unchanged. Find the acceleration of the system and the tension in the tow-bar. [4]

A constant resistance of magnitude 1400 N acts on a car of mass 1250 kg.

1. The car is moving along a straight level road at a constant speed of 28 m s\(^{-1}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car now travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.12\), with the engine working at 43.5 kW. Find this speed. [3]
3. On another occasion, the car pulls a trailer of mass 600 kg up the same hill. The system of the car and the trailer is modelled as particles connected by a light inextensible cable. The car's engine produces a driving force of 5000 N and the resistance to the motion of the trailer is 300 N. The resistance to the motion of the car remains 1400 N. Find the acceleration of the system and the tension in the cable. [4]

\includegraphics{figure_6_1} **Fig. 6.1** Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of 30° to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.

1. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane. [6]
2. \includegraphics{figure_6_2} **Fig. 6.2** It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level. [6]

\includegraphics{figure_5} A block \(A\) of mass 80 kg is connected by a light, inextensible rope to a block \(B\) of mass 40 kg. The rope joining the two blocks is taut and is parallel to a line of greatest slope of a plane which is inclined at an angle of \(20°\) to the horizontal. A force of magnitude 500 N inclined at an angle of \(15°\) above the same line of greatest slope acts on \(A\) (see diagram). The blocks move up the plane and there is a resistance force of 50 N on \(B\), but no resistance force on \(A\).

1. Find the acceleration of the blocks and the tension in the rope. [5]

1. Find the time that it takes for the blocks to reach a speed of \(1.2 \text{ m s}^{-1}\) from rest. [2]

Two particles, of masses \(1.8\) kg and \(1.2\) 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. Find the magnitude of the acceleration of the particles and find the tension in the string. [4]

\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_4} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.3 \text{ kg}\) respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. Particle \(A\) hangs freely and particle \(B\) is in contact with the table (see diagram).

1. The system is in limiting equilibrium with the string taut and \(A\) about to move downwards. Find the coefficient of friction between \(B\) and the table. [4]

A force now acts on particle \(B\). This force has a vertical component of \(1.8 \text{ N}\) upwards and a horizontal component of \(X \text{ N}\) directed away from the pulley.

1. The system is now in limiting equilibrium with the string taut and \(A\) about to move upwards. Find \(X\). [3]

\includegraphics{figure_1} A block \(B\) of mass 5 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 4 kg is attached to other end of the string. The string passes over a smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on the ground (see diagram). State the tension in the string and find the force exerted on \(B\) by the ground. [3]

\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_6} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is 3.95 m s\(^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.45 \text{ kg}\) respectively, are connected by a light inextensible string of length \(2.8 \text{ m}\). The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is \(2 \text{ m}\) above the floor. Particle \(A\) is held in contact with the surface at a distance of \(2.1 \text{ m}\) from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is \(0.3\). Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \text{ m s}^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

\includegraphics{figure_2} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a smooth horizontal surface with the string passing over a small smooth pulley fixed at the edge of the surface. \(B\) hangs vertically below the pulley at a distance \(h\) m above the floor (see diagram). \(A\) is released and the particles move. \(B\) reaches the floor and \(A\) subsequently reaches the pulley with a speed of \(3 \text{ m s}^{-1}\).

1. Explain briefly why the speed with which \(B\) reaches the floor is \(3 \text{ m s}^{-1}\). [1]
2. Find the value of \(h\). [4]