3.03k Connected particles: pulleys and equilibrium

Two objects, \(M\) and \(N\), are connected by a light inextensible string that passes over a smooth peg. \(M\) has a mass of 0.6 kilograms. \(N\) has a mass of 0.5 kilograms. \(M\) and \(N\) are initially held at rest, with the string taut, as shown in the diagram below. \includegraphics{figure_19} \(M\) and \(N\) are released at the same instant and begin to move vertically. You may assume that air resistance can be ignored.

1. It is given that \(M\) and \(N\) move with acceleration \(a\) m s\(^{-2}\) By forming two equations of motion show that $$a = \frac{1}{11}g$$ [5 marks]
2. The speed of \(N\), 0.5 seconds after its release, is \(\frac{g}{k}\) m s\(^{-1}\) where \(k\) is a constant. Find the value of \(k\) [2 marks]
3. State one assumption that must be made for the answer in part (b) to be valid. [1 mark]

In this question use \(g = 9.8\) m s⁻². A van of mass 1300 kg and a crate of mass 300 kg are connected by a light inextensible rope. The rope passes over a light smooth pulley, as shown in the diagram. The rope between the pulley and the van is horizontal. \includegraphics{figure_17} Initially, the van is at rest and the crate rests on the lower level. The rope is taut. The van moves away from the pulley to lift the crate from the lower level. The van's engine produces a constant driving force of 5000 N. A constant resistance force of magnitude 780 N acts on the van. Assume there is no resistance force acting on the crate.

1. 1. Draw a diagram to show the forces acting on the crate while it is being lifted. [1 mark]
   2. Draw a diagram to show the forces acting on the van while the crate is being lifted. [1 mark]
2. Show that the acceleration of the van is 0.80 m s⁻² [4 marks]
3. Find the tension in the rope. [2 marks]
4. Suggest how the assumption of a constant resistance force could be refined to produce a better model. [1 mark]

A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram. \includegraphics{figure_6} The combined mass of the buggy and driver is 410 kg A driving force of 300 N and a total resistance force of 140 N act on the buggy. The mass of the roller-skater is 72 kg A total resistance force of R newtons acts on the roller-skater. The buggy and the roller-skater have an acceleration of 0.2 m s\(^{-2}\)

1. 1. Find R. [3 marks]
   2. Find the tension in the rope. [3 marks]
2. State a necessary assumption that you have made. [1 mark]
3. The roller-skater releases the rope at a point A, when she reaches a speed of 6 m s\(^{-1}\) She continues to move forward, experiencing the same resistance force. The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from A.
   1. Determine whether the roller-skater will stop before reaching the stationary buggy. Fully justify your answer. [5 marks]
   2. Explain the change in motion that the driver noticed. [2 marks]

Block \(A\), of mass \(0.2\) kg, lies at rest on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, such that \(\tan \theta = \frac{7}{24}\) A light inextensible string is attached to \(A\) and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope. The other end of this string is attached to particle \(B\), of mass \(2\) kg, which is held at rest so that the string is taut, as shown in the diagram below. \includegraphics{figure_18}

1. \(B\) is released from rest so that it begins to move vertically downwards with an acceleration of \(\frac{543}{625}\) g ms\(^{-2}\) Show that the coefficient of friction between \(A\) and the surface of the inclined plane is \(0.17\) [8 marks]
2. In this question use \(g = 9.81\text{ ms}^{-2}\) When \(A\) reaches a speed of \(0.5\text{ ms}^{-1}\) the string breaks.
   1. Find the distance travelled by \(A\) after the string breaks until first coming to rest. [4 marks]
   2. State an assumption that could affect the validity of your answer to part (b)(i). [1 mark]

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]

Two particles \(A\) and \(B\) are connected by a light inextensible string of length \(1.26\) m. Particle \(A\) has a mass of \(1.25\) kg and moves on a smooth horizontal table in a circular path of radius \(0.9\) m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) has a mass of \(2\) kg and moves in a horizontal circle as shown in the diagram. The angle that the portion of string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos\theta = \frac{4}{5}\) (see diagram). \includegraphics{figure_7}

1. Determine the angular speed of \(A\) and the angular speed of \(B\). [5]

At the start of the motion, \(A\), \(O\) and \(B\) all lie in the same vertical plane.

1. Find the first subsequent time when \(A\), \(O\) and \(B\) all lie in the same vertical plane. [2]

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]

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]

A box \(A\) of mass 0.8 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a smooth pulley fixed at the edge of the table. The other end of the string is attached to a sphere \(B\) of mass 1.2 kg, which hangs freely below the pulley. The magnitude of the frictional force between \(A\) and the table is \(F\) N. The system is released from rest when the string is taut. After release, \(B\) descends a distance of 0.9 m in 0.8 s. Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\), [2]
2. the tension in the string, [3]
3. the value of \(F\). [3]

A car of mass \(1200 \text{ kg}\) pulls a trailer of mass \(400 \text{ kg}\) along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of \(3200 \text{ N}\). The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is \(0.4 \text{ m s}^{-2}\),

1. find the resistance to motion on the trailer, [4]
2. find the tension in the tow-rope. [3]

When the car and trailer are travelling at \(25 \text{ m s}^{-1}\) the tow-rope breaks. Assuming that the resistances to motion remain unchanged,

1. find the distance the trailer travels before coming to a stop, [4]
2. state how you have used the modelling assumption that the tow-rope is inextensible. [1]

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]

Two particles, \(A\) and \(B\), each of mass 1 kg are connected by a light inextensible string. Particle \(A\) is at rest on a slope inclined at 30° to the horizontal. The string passes over a small smooth pulley at the top of the slope and particle \(B\) hangs freely, as shown in the diagram. \includegraphics{figure_11}

1. 1. In the case when the slope is smooth, draw a fully labelled diagram to show the forces acting on the particles. Hence find the acceleration of the particles and the tension in the string. [7]
   2. Write down the direction of the resultant force exerted by the string on the pulley. [1]
2. In fact the contact between particle \(A\) and the slope is rough. The coefficient of friction between \(A\) and the slope is \(\mu\). The system is in equilibrium. Find the set of possible values of \(\mu\). [5]

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]

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{figure_8} The acceleration of the system is 0.3 m s\(^{-2}\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). [2]

Truck \(S\) is now subjected to an extra resistive force of 1800 N. The pulling force, \(P\), does not change.

1. Calculate the new acceleration of the trucks. [2]
2. Calculate the force in the coupling between the trucks. [2]

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{figure_8} The acceleration of the system is 0.3 ms\(^{-2}\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). [2]

Truck \(S\) is now subjected to an extra resistive force of 1800 N. The pulling force, \(P\), does not change.

1. Calculate the new acceleration of the trucks. [2]
2. Calculate the force in the coupling between the trucks. [2]

A particle \(P\) of mass \(2\) kg rests on a long rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.2\). A light inextensible string has one end attached to \(P\) and the other end attached to a particle \(Q\) of mass \(4\) kg. The particle \(Q\) is placed on a smooth plane inclined at \(30^{\circ}\) to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time \(t = 0\), where \(t\) is measured in seconds. In the subsequent motion \(P\) does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(\frac{8}{3}\) m s\(^{-2}\). [4]

After the particles have moved a distance of \(12\) m the string is cut.

1. Find the corresponding value of \(t\) and the speed of the particles at this instant. [4]
2. Find the value of \(t\) when \(P\) comes to rest. [3]