3.03k Connected particles: pulleys and equilibrium

7. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(5 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). The string passes over a small, smooth, light fixed pulley. Particle \(A\) is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. Particle A is released.

1. Find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the pulley by the string while \(A\) is falling and before \(B\) hits the pulley.
2. State how, in your solution to part (a), you have used the fact that the pulley is smooth.

3. A tractor of mass 6 tonnes is dragging a large block of mass 2 tonnes along rough horizontal ground. The cable connecting the tractor to the block is horizontal and parallel to the direction of motion. The cable is modelled as being light and inextensible.
The driving force of the tractor is 7400 N and the resistance to the motion of the tractor is 200 N . The resistance to the motion of the block is \(R\) newtons, where \(R\) is a constant. Given that the tension in the cable is 6000 N and the tractor is accelerating,

1. find the value of \(R\).
2. State how you have used the fact that the cable is modelled as being inextensible.
   

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3. \begin{figure}[h] \end{figure} Two particles, \(P\) and \(Q\), have masses \(4 m\) and \(2 m\) respectively. The particles are connected by a light inextensible string. A second light inextensible string has one end attached to \(Q\). Both strings are taut and vertical, as shown in Figure 1. The particles are accelerating vertically downwards.
Given that the tension in the string connecting the two particles is \(3 m g\), find, in terms of \(m\) and \(g\), the tension in the upper string.

8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(2 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). Particle \(A\) is held at rest on a rough plane which is inclined to horizontal ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. Particle \(B\) hangs vertically below \(P\) with the string taut, at a height \(h\) above the ground, as shown in Figure 4. The part of the string between \(A\) and \(P\) lies along a line of greatest slope of the plane. The two particles, the string and the pulley all lie in the same vertical plane.
The coefficient of friction between \(A\) and the plane is \(\frac { 11 } { 36 }\) The particle \(A\) is released from rest and begins to move up the plane.

1. Show that the frictional force acting on \(A\) as it moves up the plane is \(\frac { 22 m g } { 39 }\)
2. Write down an equation of motion for \(B\).
3. Show that the acceleration of \(A\) immediately after its release is \(\frac { 1 } { 3 } g\) In the subsequent motion, \(A\) comes to rest before it reaches the pulley.
4. Find, in terms of \(h\), the total distance travelled by \(A\) from when it was released from rest to when it first comes to rest again.

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8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\). Initially \(P\) is held at rest on the inclined plane with the part of the string from \(P\) to the pulley parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between \(P\) and the plane is \(\mu\). The system is released from rest, with the string taut, and \(Q\) strikes the ground before \(P\) reaches the pulley. The speed of \(Q\) at the instant when it strikes the ground is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion before \(Q\) strikes the ground, find the tension in the string.
2. Find the value of \(\mu\).

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7. A truck of mass 1600 kg is towing a car of mass 960 kg along a straight horizontal road. The truck and the car are joined by a light rigid tow bar. The tow bar is horizontal and is parallel to the direction of motion. The truck and the car experience constant resistances to motion of magnitude 640 N and \(R\) newtons respectively. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitude of the acceleration of the truck and the car is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 436\)
2. Find the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 15 }\). The truck and the car move down a line of greatest slope of the hill with the tow bar parallel to the direction of motion. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitudes of the resistances to motion on the truck and the car are 640 N and 436 N respectively.
3. Find the magnitude of the acceleration of the truck and the car as they move down the hill.
   
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7. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(2 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). The string passes over a small, smooth, light pulley \(P\) which is fixed at the top of a rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) Particle \(A\) is held at rest on the plane with the string taut and \(B\) hanging freely below \(P\), as shown in Figure 4. The section of the string \(A P\) is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 2 }\) Particle \(A\) is released and begins to move up the plane.
For the motion before \(A\) reaches the pulley,

1. 1. write down an equation of motion for \(A\),
   2. write down an equation of motion for \(B\),
2. find, in terms of \(g\), the acceleration of \(A\),
3. find the magnitude of the force exerted on the pulley by the string.
4. State how you have used the information that \(P\) is a smooth pulley.

7. \begin{figure}[h] \end{figure} Figure 4 shows a block \(A\) of mass \(m\) held at rest on a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal and the coefficient of friction between the block and the plane is \(\mu\). One end of a light inextensible string is now attached to \(A\). The string passes over a small smooth pulley which is fixed at the top of the plane.
The other end of the string is attached to a block \(B\) of mass \(k m\).
Block \(B\) hangs vertically below the pulley, with the string taut.
The string from \(A\) to the pulley lies along a line of greatest slope of the plane.
Both \(A\) and \(B\) are modelled as particles.
When the system is released from rest, \(A\) moves up the plane and the tension in the string is \(\frac { 4 m g } { 3 }\) Given that \(\mu = \frac { 1 } { 3 }\) and \(\tan \alpha = \frac { 7 } { 24 }\)

1. 1. find the magnitude of the acceleration of \(A\), giving your answer in terms of \(g\),
   2. find the value of \(k\).
2. Find the magnitude of the resultant force exerted on the pulley by the string, giving your answer in terms of \(m\) and \(g\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac { 4 } { 3 }\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,

1. find the value of \(m\),
2. find the magnitude of the force exerted on the pulley by the string,
3. find the direction of the force exerted on the pulley by the string. \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-21_2258_50_314_37}
   

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3. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(3 m\) and \(k m\) respectively, where \(k > 3\). They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown in Fig. 3. While the particles are moving freely, \(A\) has an acceleration of magnitude \(\frac { 2 } { 5 } g\).

1. Find, in terms of \(m\) and g , the tension in the string.
2. State why \(B\) also has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
3. Find the value of \(k\).
4. State how you have used the fact that the string is light.

7. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a block \(P\) of mass 5 kg . The block \(P\) is held at rest on a smooth fixed plane which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The string lies along a line of greatest slope of the plane and passes over a smooth light pulley which is fixed at the top of the plane. The other end of the string is attached to a light scale pan which carries two blocks \(Q\) and \(R\), with block \(Q\) on top of block \(R\), as shown in Figure 3. The mass of block \(Q\) is 5 kg and the mass of block \(R\) is 10 kg . The scale pan hangs at rest and the system is released from rest. By modelling the blocks as particles, ignoring air resistance and assuming the motion is uninterrupted, find

1. 1. the acceleration of the scale pan,
   2. the tension in the string,
2. the magnitude of the force exerted on block \(Q\) by block \(R\),
3. the magnitude of the force exerted on the pulley by the string.

5 A car is towing a trailer along a straight road using a light tow-bar which is parallel to the road. The masses of the car and the trailer are 900 kg and 250 kg respectively. The resistance to motion of the car is 600 N and the resistance to motion of the trailer is 150 N .

1. At one stage of the motion, the road is horizontal and the pulling force exerted on the trailer is zero.
   1. Show that the acceleration of the trailer is \(- 0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the driving force exerted by the car.
   3. Calculate the distance required to reduce the speed of the car and trailer from \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   4. At another stage of the motion, the car and trailer are moving down a slope inclined at \(3 ^ { \circ }\) to the horizontal. The resistances to motion of the car and trailer are unchanged. The driving force exerted by the car is 980 N . Find
      (a) the acceleration of the car and trailer,
      (b) the pulling force exerted on the trailer.

7 \includegraphics[max width=\textwidth, alt={}, center]{db77a63a-6ff8-4fe5-bdd0-15afb7eb4866-4_419_419_274_735} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are released from rest, with the string taut, and \(A\) and \(B\) at the same height above a horizontal floor (see diagram). In the subsequent motion, \(A\) descends with acceleration \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and strikes the floor 0.8 s after being released. It is given that \(B\) never reaches the pulley.

1. Calculate the distance \(A\) moves before it reaches the floor and the speed of \(A\) immediately before it strikes the floor.
2. Show that \(B\) rises a further 0.064 m after \(A\) strikes the floor, and calculate the total length of time during which \(B\) is rising.
3. Sketch the ( \(t , v\) ) graph for the motion of \(B\) from the instant it is released from rest until it reaches a position of instantaneous rest.
4. Before \(A\) strikes the floor the tension in the string is 5.88 N . Calculate the mass of \(A\) and the mass of \(B\).
5. The pulley has mass 0.5 kg , and is held in a fixed position by a light vertical chain. Calculate the tension in the chain
   1. immediately before \(A\) strikes the floor,
   2. immediately after \(A\) strikes the floor.

2 A trailer of mass 500 kg is attached to a car of mass 1250 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road. The resistance to motion of the trailer is 400 N and the resistance to motion of the car is 900 N . Find both the tension in the tow-bar and the driving force of the car in each of the following cases.

1. The car and trailer are travelling at constant speed.
2. The car and trailer have acceleration \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

2 \includegraphics[max width=\textwidth, alt={}, center]{99d30766-9c1b-43a8-986a-112b78b08146-2_643_289_1475_927} Particles \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is 0.4 N and the air resistance acting on \(B\) is 0.25 N (see diagram). The downward acceleration of each of the particles is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the string is \(T \mathrm {~N}\).

1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\).
2. Find the values of \(a\) and \(T\).

6 A block \(B\) of mass 0.85 kg lies on a smooth slope inclined at \(30 ^ { \circ }\) to the horizontal. \(B\) is attached to one end of a light inextensible string which is parallel to the slope. At the top of the slope, the string passes over a smooth pulley. The other end of the string hangs vertically and is attached to a particle \(P\) of mass 0.55 kg . The string is taut at the instant when \(P\) is projected vertically downwards.

1. Calculate
   1. the acceleration of \(B\) and the tension in the string,
   2. the magnitude of the force exerted by the string on the pulley. The initial speed of \(P\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after moving \(1.5 \mathrm {~m} P\) reaches the ground, where it remains at rest. \(B\) continues to move up the slope and does not reach the pulley.
   3. Calculate the total distance \(B\) moves up the slope before coming instantaneously to rest.

2 Particles of mass 2 kg and 4 kg are attached to the ends \(X\) and \(Y\) of a light, inextensible string. The string passes round fixed, smooth pulleys at \(\mathrm { P } , \mathrm { Q }\) and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h] \end{figure}

1. State what information in the question tells you that
   (A) the tension is the same throughout the string,
   (B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is \(T \mathrm {~N}\) and the magnitude of the acceleration of the particles is \(a \mathrm {~ms} ^ { - 2 }\).
2. Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
3. Write down equations of motion for the particles at X and at Y . Hence calculate the values of \(T\) and \(a\).

2 Fig. 2 shows a light string with an object of mass 4 kg attached at end A . The string passes over a smooth pulley and its other end B is attached to two light strings BC and BD of the same length. The strings BC and BD are attached to horizontal ground and are each inclined at \(20 ^ { \circ }\) to the vertical. The system is in equilibrium. \begin{figure}[h] \end{figure}

1. What information in the question tells you that the tension is the same throughout the string AB ?
2. What is the tension in the string AB ?
3. Calculate the tension in the strings BC and BD .

4 A car and its trailer travel along a straight, horizontal road. The coupling between them is light and horizontal. The car has mass 900 kg and resistance to motion 100 N , the trailer has mass 700 kg and resistance to motion 300 N , as shown in Fig. 4. The car and trailer have an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h] \end{figure}

1. Calculate the driving force of the car.
2. Calculate the force in the coupling.

5 A block of mass 4 kg slides on a horizontal plane against a constant resistance of 14.8 N . A light, inextensible string is attached to the block and, after passing over a smooth pulley, is attached to a freely hanging sphere of mass 2 kg . The part of the string between the block and the pulley is horizontal. This situation is shown in Fig. 5. \begin{figure}[h] \end{figure} The tension in the string is \(T \mathrm {~N}\) and the acceleration of the block and of the sphere is \(a \mathrm {~ms} ^ { - 2 }\).

1. Write down the equation of motion of the block and also the equation of motion of the sphere, each in terms of \(T\) and \(a\).
2. Find the values of \(T\) and \(a\).

5 Fig. 5 shows two boxes, A of mass 12 kg and B of mass 6 kg , sliding in a straight line on a rough horizontal plane. The boxes are connected by a light rigid rod which is parallel to the line of motion. The only forces acting on the boxes in the line of motion are those due to the rod and a constant force of \(F \mathrm {~N}\) on each box. \begin{figure}[h] \end{figure} The boxes have an initial speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and come to rest after sliding a distance of 0.375 m .

1. Calculate the deceleration of the boxes and the value of \(F\).
2. Calculate the magnitude of the force in the rod and state, with a reason, whether it is a tension or a thrust (compression).

1 Fig. 1 shows two blocks of masses 3 kg and 5 kg connected by a light string which passes over a smooth, fixed pulley. Initially the blocks are held at rest but then they are released. \begin{figure}[h] \end{figure} Find the acceleration of the blocks when they start to move, and the tension in the string.

3 Fig. 3 shows a system in equilibrium. The rod is firmly attached to the floor and also to an object, P. The light string is attached to P and passes over a smooth pulley with an object Q hanging freely from its other end. \begin{figure}[h] \end{figure}

1. Why is the tension the same throughout the string?
2. Calculate the force in the rod, stating whether it is a tension or a thrust.

2 Fig. 2 shows a 6 kg block on a smooth horizontal table. It is connected to blocks of mass 2 kg and 9 kg by two light strings which pass over smooth pulleys at the edges of the table. The parts of the strings attached to the 6 kg block are horizontal. \begin{figure}[h] \end{figure}

1. Draw three separate diagrams showing all the forces acting on each of the blocks.
2. Calculate the acceleration of the system and the tension in each string.

3 Fig. 3.1 shows a block of mass 8 kg on a smooth horizontal table.
This block is connected by a light string passing over a smooth pulley to a block of mass 4 kg which hangs freely. The part of the string between the 8 kg block and the pulley is parallel to the table. The system has acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h] \end{figure}

1. Write down two equations of motion, one for each block.
2. Find the value of \(a\). The table is now tilted at an angle of \(\theta\) to the horizontal as shown in Fig. 3.2. The system is set up as before; the 4 kg block still hangs freely. \begin{figure}[h]
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3. The system is now in equilibrium. Find the value of \(\theta\).