3.03l Newton's third law: extend to situations requiring force resolution

\includegraphics{figure_7} The diagram shows a triangular block with sloping faces inclined to the horizontal at \(45°\) and \(30°\). Particle \(A\) of mass \(0.8 \text{ kg}\) lies on the face inclined at \(45°\) and particle \(B\) of mass \(1.2 \text{ kg}\) lies on the face inclined at \(30°\). The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the faces. The parts \(AP\) and \(BP\) of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.

1. Given that both faces are smooth, find the speed of \(A\) after each particle has travelled a distance of \(0.4 \text{ m}\). [6]
2. It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is \(\mu\). Find the value of \(\mu\) for which the system is in limiting equilibrium. [6]

\includegraphics{figure_7} A rough inclined plane of length 65 cm is fixed with one end at a height of 16 cm above the other end. Particles \(P\) and \(Q\), of masses \(0.13\) kg and \(0.11\) kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley at the top of the plane. Particle \(P\) is held at rest on the plane and particle \(Q\) hangs vertically below the pulley (see diagram). The system is released from rest and \(P\) starts to move up the plane.

1. Draw a diagram showing the forces acting on \(P\) during its motion up the plane. [1]
2. Show that \(T - F > 0.32\), where \(T\) N is the tension in the string and \(F\) N is the magnitude of the frictional force on \(P\). [4]

The coefficient of friction between \(P\) and the plane is 0.6.

1. Find the acceleration of \(P\). [6]

\includegraphics{figure_6} Two particles \(P\) and \(Q\), each of mass \(m\) kg, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The plane is inclined at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). Particle \(P\) rests on the plane and particle \(Q\) hangs vertically, as shown in the diagram. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium.

1. Show that the coefficient of friction between \(P\) and the plane is \(\frac{4}{3}\). [5]

A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at 2.5 m s\(^{-2}\).

1. Find the value of \(m\). [5]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of masses \(0.4\) kg and \(0.7\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The coefficient of friction between \(P\) and the plane is \(0.5\). The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) lies on the plane and particle \(Q\) hangs vertically. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). A force of magnitude \(X\) N, acting directly down the plane, is applied to \(P\).

1. Show that the greatest value of \(X\) for which \(P\) remains stationary is \(6.2\). [4]
2. Given instead that \(X = 0.8\), find the acceleration of \(P\). [4]

\includegraphics{figure_2} Figure 2 shows two particles \(A\) and \(B\), of masses \(3m\) and \(4m\) respectively, attached to the ends of a light inextensible string. Initially \(A\) is held at rest on the surface of a fixed rough inclined plane. 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 light 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. The particle \(B\) hangs freely and is vertically below \(P\). The system is released from rest with the string taut and with \(B\) at a height of 1.75 m above the ground. In the subsequent motion, \(A\) does not hit the pulley. For the period before \(B\) hits the ground,

1. write down an equation of motion for each particle. [4]
2. Hence show that the acceleration of \(B\) is \(\frac{8}{35}g\). [5]
3. Explain how you have used the fact that the string is inextensible in your calculation. [1]

When \(B\) hits the ground, \(B\) does not rebound and comes immediately to rest.

1. Find the distance travelled by \(A\) from the instant when the system is released to the instant when \(A\) first comes to rest. [7]

\includegraphics{figure_4} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. Another particle \(Q\), also of mass \(m\), is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle \(Q\) is held at rest on the table and particle \(P\) hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, \(P\) and \(Q\) all lie in the same vertical plane. The coefficient of friction between \(Q\) and the table is \(\mu\), where \(\mu < 1\) Particle \(Q\) is released from rest. The tension in the string before \(Q\) hits the pulley is \(kmg\), where \(k\) is a constant.

1. Find \(k\) in terms of \(\mu\). [7] Given that \(Q\) is initially a distance \(d\) from the pulley,
2. find, in terms of \(d\), \(g\) and \(\mu\), the time taken by \(Q\), after release, to reach the pulley. [4]
3. Describe what would happen if \(\mu \geqslant 1\), giving a reason for your answer. [2]

\includegraphics{figure_3} Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley.

1. Find the tension in the string immediately after the particles are released. [6]
2. Find the acceleration of \(A\) immediately after the particles are released. [2]

When the particles have been moving for 0.5 s, the string breaks.

1. Find the further time that elapses until \(B\) hits the floor. [9]

\includegraphics{figure_4} Two particles \(P\) and \(Q\) have masses \(3m\) and \(5m\) respectively. They are connected by a light inextensible string which passes over a small smooth light pulley fixed at the edge of a rough horizontal table. Particle \(P\) lies on the table and particle \(Q\) hangs freely below the pulley, as shown in Fig. 4. The coefficient of friction between \(P\) and the table is 0.6. The system is released from rest with the string taut. For the period before \(Q\) hits the floor or \(P\) reaches the pulley,

1. write down an equation of motion for each particle separately, [4]
2. find, in terms of \(g\), the acceleration of \(Q\), [4]
3. find, in terms of \(m\) and \(g\), the tension in the string. [2]

When \(Q\) has moved a distance \(h\), it hits the floor and the string becomes slack. Given that \(P\) remains on the table during the subsequent motion and does not reach the pulley,

1. find, in terms of \(h\), the distance moved by \(P\) after the string becomes slack until \(P\) comes to rest. [6]

\includegraphics{figure_4} A particle \(A\) of mass 0.8 kg rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass 1.2 kg which hangs freely below the pulley, as shown in Fig. 4. The system is released from rest with the string taut and with \(B\) at a height of 0.6 m above the ground. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground. In an initial model of the situation, the table is assumed to be smooth. Using this model, find

1. the tension in the string before \(B\) reaches the ground, [5]
2. the time taken by \(B\) to reach the ground. [3]

In a refinement of the model, it is assumed that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac{1}{4}\). Using this refined model,

1. find the time taken by \(B\) to reach the ground. [8]

\includegraphics{figure_4} Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15°\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find

1. the acceleration of the lorry and the car, [3]
2. the tension in the towbar. [4]

When the speed of the vehicles is \(6 \text{ m s}^{-1}\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N,

1. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. [4]
2. State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer. [2]

A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find

1. the acceleration of the car and trailer, [3]
2. the magnitude of the tension in the towbar. [3]

The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude \(F\) newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

1. find the value of \(F\). [7]

\includegraphics{figure_3} Figure 3 shows two particles \(A\) and \(B\), of mass \(m\) kg and 0.4 kg respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\). The system is released from rest with the string taut and \(B\) descends with acceleration \(\frac{1}{8}g\).

1. Write down an equation of motion for \(B\). [2]
2. Find the tension in the string. [2]
3. Prove that \(m = \frac{16}{35}\). [4]
4. State where in the calculations you have used the information that \(P\) is a light smooth pulley. [1]

On release, \(B\) is at a height of one metre above the ground and \(AP = 1.4\) m. The particle \(B\) strikes the ground and does not rebound.

1. Calculate the speed of \(B\) as it reaches the ground. [2]
2. Show that \(A\) comes to rest as it reaches \(P\). [5]

END

\includegraphics{figure_1} The particles have mass 3 kg and \(m\) kg, where \(m < 3\). They are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are held in position with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The particles are then released from rest. The initial acceleration of each particle has magnitude \(\frac{1}{2}g\). Find

1. the tension in the string immediately after the particles are released, [3]
2. the value of \(m\). [4]

Two particles \(P\) and \(Q\), of masses \(3\) kg and \(2\) kg respectively, rest on the smooth faces of a wedge whose cross-section is a triangle with angles \(30°\), \(60°\) and \(90°\), as shown. \(P\) and \(Q\) are connected by a light string, parallel to the lines of greatest slope of the two planes, which passes over a fixed pulley at the highest point of the wedge. \includegraphics{figure_6} The system is released from rest with \(P\) \(0.8\) m from the pulley and \(Q\) \(1\) m from the bottom of the wedge, and \(Q\) starts to move down. Calculate

1. the acceleration of either particle, \hfill [5 marks]
2. the tension in the string, \hfill [2 marks]
3. the speed with which \(P\) reaches the pulley. \hfill [3 marks]

Two modelling assumptions have been made about the string and the pulley.

1. State these two assumptions and briefly describe how you have used each one in your solution. \hfill [4 marks]

Two particles \(P\) and \(Q\), of masses \(2m\) and \(3m\) respectively, are connected by a light string. Initially, \(P\) is at rest on a smooth horizontal table. The string passes over a small smooth pulley and \(Q\) rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{4}{3}\). The coefficient of friction between \(Q\) and the inclined plane is \(\frac{1}{6}\). \includegraphics{figure_7} The system is released from rest with \(Q\) at a distance of 0.8 metres above a horizontal floor.

1. Show that the acceleration of \(P\) and \(Q\) is \(\frac{21g}{50}\), stating a modelling assumption which you must make to ensure that both particles have the same acceleration. [7 marks]
2. Find the speed with which \(Q\) hits the floor. [2 marks]

After \(Q\) hits the floor and does not rebound, \(P\) moves a further 0.2 m until it hits the pulley.

1. Find the total time after the system is released before \(P\) hits the pulley. [5 marks]

\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_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]

Two heavy boxes, \(M\) and \(N\), are connected securely by a length of rope. The mass of \(M\) is 50 kilograms. The mass of \(N\) is 80 kilograms. \(M\) is placed near the bottom of a rough slope. The slope is inclined at 60° above the horizontal. The rope is passed over a smooth pulley at the top end of the slope so that \(N\) hangs with the rope vertical. The boxes are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below. \includegraphics{figure_21} When the boxes are released, \(M\) moves up the slope as \(N\) descends, with acceleration \(a\) m s\(^{-2}\) The tension in the rope is \(T\) newtons.

1. Explain why the equation of motion for \(N\) is $$80g - T = 80a$$ [1 mark]
2. Show that the normal reaction force between \(M\) and the slope is \(25g\) newtons. [1 mark]
3. The coefficient of friction, \(\mu\), between the slope and \(M\) is such that \(0 \leq \mu \leq 1\) Show that $$a \geq \frac{(11 - 5\sqrt{3})g}{26}$$ [6 marks]
4. State one modelling assumption you have made throughout this question. [1 mark]

The diagram below shows two particles \(A\) and \(B\), of mass 2 kg and 5 kg respectively, which are connected by a light inextensible string passing over a fixed smooth pulley. Initially, \(B\) is held at rest with the string just taut. It is then released. \includegraphics{figure_4}

1. Calculate the magnitude of the acceleration of \(A\) and the tension in the string. [6]
2. What assumption does the word 'light' in the description of the string enable you to make in your solution? [1]

A truck of mass 180 kg runs on smooth horizontal rails. A light inextensible rope is attached to the front of the truck. The rope runs parallel to the rails until it passes over a light smooth pulley. The rest of the rope hangs down a vertical shaft. When the truck is required to move, a load of \(M\) kg is attached to the end of the rope in the shaft and the brakes are then released.

1. Find the tension in the rope when the truck and the load move with an acceleration of magnitude 0.8 ms\(^{-2}\) and calculate the corresponding value of \(M\). [5]
2. In addition to the assumptions given in the question, write down one further assumption that you have made in your solution to this problem and explain how that assumption affects both of your answers. [3]

\includegraphics{figure_7} A light inextensible string of length 8 m is threaded through a smooth fixed ring, \(R\), and carries a particle at each end. One particle, \(P\), of mass 0.5 kg is at rest at a distance 3 m below \(R\). The other particle, \(Q\), is rotating in a horizontal circle whose centre coincides with the position of \(P\) (see diagram). Find the angular speed and the mass of \(Q\). [8]