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

6. \begin{figure}[h] \end{figure} A railway engine of mass 1500 kg is attached to a railway truck of mass 500 kg by a straight rigid coupling. The engine pushes the truck up a straight track, which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 7 } { 25 }\). The coupling is parallel to the track and parallel to the direction of motion, as shown in Figure 3. The engine produces a constant driving force of magnitude \(D\) newtons. The engine and the truck experience constant resistances to motion, from non-gravitational forces, of magnitude 1200 N and 500 N respectively. The thrust in the coupling is 2000 N . The coupling is modelled as a light rod.

1. Find the acceleration of the engine and the truck.
2. Find the value of \(D\).

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 \(\mu\).

   END

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.
   
   \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-27_67_59_2654_1886}

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 \(g\).

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} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough horizontal table. The string passes over a small smooth pulley \(P\) fixed on the edge of the table. The particle \(B\) hangs freely below the pulley, as shown in Figure 3. The coefficient of friction between \(A\) and the table is \(\mu\). The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of \(A\) and \(B\) is \(\frac { 4 } { 9 } g\). By writing down separate equations of motion for \(A\) and \(B\),

1. find the tension in the string immediately after the particles begin to move,
2. show that \(\mu = \frac { 2 } { 3 }\). When \(B\) has fallen a distance \(h\), it hits the ground and does not rebound. Particle \(A\) is then a distance \(\frac { 1 } { 3 } h\) from \(P\).
3. Find the speed of \(A\) as it reaches \(P\).
4. State how you have used the information 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.

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 \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

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.

5. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 4. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 250 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 150 N . The towbar is modelled as a light rod.
At the instant when the speed of the van is \(16 \mathrm {~ms} ^ { - 1 }\), the engine of the van is working at a rate of 10 kW .

1. Find the deceleration of the van at this instant.
2. Find the tension in the towbar at this instant.

2. A trailer of mass 250 kg is towed by a car of mass 1000 kg . The car and the trailer are travelling down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) The resistance to motion of the car is modelled as a single force of magnitude 300 N acting parallel to the road. The resistance to motion of the trailer is modelled as a single force of magnitude 100 N acting parallel to the road. The towbar joining the car to the trailer is modelled as a light rod which is parallel to the direction of motion. At a given instant the car and the trailer are moving down the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the power being developed by the car's engine at this instant.
2. Find the tension in the towbar at this instant.

3. Two particles \(P\) and \(Q\), of mass \(2 m\) and \(3 m\) respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed rough plane inclined at \(\theta\) to the horizontal ground, where \(\sin \theta = \frac { 2 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. At time \(t = 0\) the system is released from rest with the string taut. When \(P\) moves the friction between \(P\) and the plane is modelled as a constant force of magnitude \(\frac { 3 } { 5 } m g\). At the instant when each particle has moved a distance \(d\), they are both moving with speed \(v\), particle \(P\) has not reached the pulley and \(Q\) has not reached the ground.

1. Show that the total potential energy lost by the system when each particle has moved a distance \(d\) is \(\frac { 11 } { 5 } m g d\).
2. Use the work-energy principle to find \(v ^ { 2 }\) in terms of \(g\) and \(d\). When \(t = T\) seconds, \(d = 1.5 \mathrm {~m}\).
3. Find the value of \(T\).
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7 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-4_369_508_246_781} Particles \(P\) and \(Q\), of masses \(m \mathrm {~kg}\) and 0.05 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth pulley. \(Q\) is attached to a particle \(R\) of mass 0.45 kg by a light inextensible string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. \(P\) is in contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the string \(Q R\) is 2.52 N during the descent of \(R\).

1. (a) Find the acceleration of \(R\) during its descent.
   (b) By considering the motion of \(Q\), calculate the tension in the string \(P Q\) during the descent of \(R\).
2. Find the value of \(m\). \(R\) strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, \(P\) does not reach the pulley and \(Q\) does not reach the surface.
3. Calculate the greatest height of \(P\) above the surface.

5 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-3_291_182_799_945} Particles \(P\) and \(Q\), of masses 0.4 kg and \(m \mathrm {~kg}\) respectively, are joined by a light inextensible string which passes over a smooth pulley. The particles are released from rest at the same height above a horizontal surface; the string is taut and the portions of the string not in contact with the pulley are vertical (see diagram). \(Q\) begins to descend with acceleration \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and reaches the surface 0.3 s after being released. Subsequently, \(Q\) remains at rest and \(P\) never reaches the pulley.

1. Calculate the tension in the string while \(Q\) is in motion.
2. Calculate the momentum lost by \(Q\) when it reaches the surface.
3. Calculate the greatest height of \(P\) above the surface. \section*{[Questions 6 and 7 are printed overleaf.]}

7 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-4_310_579_255_721} A block \(B\) is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.6 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth pulley fixed to the top of the plane. A particle \(Q\) of mass 0.5 kg is attached to the other end of the string. The portion of the string attached to \(P\) is parallel to a line of greatest slope of the plane, the portion of the string attached to \(Q\) is vertical and the string is taut. The particles are released from rest and start to move with acceleration \(1.4 \mathrm {~ms} ^ { - 2 }\) (see diagram). It is given that \(B\) is in equilibrium while \(P\) moves on its upper surface.

1. Find the tension in the string while \(P\) and \(B\) are in contact.
2. Calculate the coefficient of friction between \(P\) and \(B\).
3. Given that the weight of \(B\) is 7 N , calculate the set of possible values of the coefficient of friction between \(B\) and the plane.

7 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-4_392_1192_255_424} \(A B\) and \(B C\) are lines of greatest slope on a fixed triangular prism, and \(M\) is the mid-point of \(B C . A B\) and \(B C\) are inclined at \(30 ^ { \circ }\) to the horizontal. The surface of the prism is smooth between \(A\) and \(B\), and between \(B\) and \(M\). Between \(M\) and \(C\) the surface of the prism is rough. A small smooth pulley is fixed to the prism at \(B\). A light inextensible string passes over the pulley. Particle \(P\) of mass 0.3 kg is fixed to one end of the string, and is placed at \(A\). Particle \(Q\) of mass 0.4 kg is fixed to the other end of the string and is placed next to the pulley on \(B C\). The particles are released from rest with the string taut. \(P\) begins to move towards the pulley, and \(Q\) begins to move towards \(M\) (see diagram).

1. Show that the initial acceleration of the particles is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and find the tension in the string. The particle \(Q\) reaches \(M 1.8 \mathrm {~s}\) after being released from rest.
2. Find the speed of the particles when \(Q\) reaches \(M\). After \(Q\) passes through \(M\), the string remains taut and the particles decelerate uniformly. \(Q\) comes to rest between \(M\) and \(C 1.4 \mathrm {~s}\) after passing through \(M\).
3. Find the deceleration of the particles while \(Q\) is moving from \(M\) towards \(C\).
4. (a) By considering the motion of \(P\), find the tension in the string while \(Q\) is moving from \(M\) towards \(C\).
   (b) Calculate the magnitude of the frictional force which acts on \(Q\) while it is moving from \(M\) towards \(C\). \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

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.
   (3 marks)
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.
   (1 mark)

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. \begin{figure}[h] \end{figure} One end of a string is attached to a small ball \(P\) of mass \(4 m\).
The other end of the string is attached to another small ball \(Q\) of mass \(3 m\).
The string passes over a fixed pulley.
Ball \(P\) is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. Ball \(P\) is released.
The string is modelled as being light and inextensible, the balls are modelled as particles, the pulley is modelled as being smooth and air resistance is ignored.

1. Using the model, find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the pulley by the string while \(P\) is falling and before \(Q\) hits the pulley.
2. State one limitation of the model, apart from ignoring air resistance, that will affect the accuracy of your answer to part (a).

4. \begin{figure}[h] \end{figure} A car of mass 1200 kg is towing a trailer of mass 400 kg along a straight horizontal road using a tow rope, as shown in Figure 2.
The rope is horizontal and parallel to the direction of motion of the car.

• The resistance to motion of the car is modelled as a constant force of magnitude \(2 R\) newtons
• The resistance to motion of the trailer is modelled as a constant force of magnitude \(R\) newtons
• The rope is modelled as being light and inextensible
• The acceleration of the car is modelled as \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

The driving force of the engine of the car is 7400 N and the tension in the tow rope is 2400 N . Using the model,

1. find the value of \(a\) In a refined model, the rope is modelled as having mass and the acceleration of the car is found to be \(a _ { 1 } \mathrm {~ms} ^ { - 2 }\)
2. State how the value of \(a _ { 1 }\) compares with the value of \(a\)
3. State one limitation of the model used for the resistance to motion of the car.

4. \begin{figure}[h] \end{figure} Figure 2 shows a car towing a trailer along a straight horizontal road.
The mass of the car is 800 kg and the mass of the trailer is 600 kg .
The trailer is attached to the car by a towbar which is parallel to the road and parallel to the direction of motion of the car and the trailer. The towbar is modelled as a light rod.
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer is modelled as a constant force of magnitude R newtons. The engine of the car is producing a constant driving force that is horizontal and of magnitude 1740 N. The acceleration of the car is \(0.6 \mathrm {~ms} ^ { - 2 }\) and the tension in the towbar is T newtons.
Using the model,

1. show that \(\mathrm { R } = 500\)
2. find the value of T . At the instant when the speed of the car and the trailer is \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks.
   The trailer moves a further distance d metres before coming to rest.
   The resistance to the motion of the trailer is modelled as a constant force of magnitude 500 N. Using the model,
3. show that, after the towbar breaks, the deceleration of the trailer is \(\frac { 5 } { 6 } \mathrm {~ms} ^ { - 2 }\)
4. find the value of d. In reality, the distance d metres is likely to be different from the answer found in part (d).
5. Give two different reasons why this is the case.

3. \begin{figure}[h] \end{figure} A ball \(P\) of mass \(2 m\) is attached to one end of a string.
The other end of the string is attached to a ball \(Q\) of mass \(5 m\).
The string passes over a fixed pulley.
The system is held at rest with the balls hanging freely and the string taut.
The hanging parts of the string are vertical with \(P\) at a height \(2 h\) above horizontal ground and with \(Q\) at a height \(h\) above the ground, as shown in Figure 1. The system is released from rest.
In the subsequent motion, \(Q\) does not rebound when it hits the ground and \(P\) does not hit 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.
Air resistance is modelled as being negligible.
Using this model,

1. 1. write down an equation of motion for \(P\),
   2. write down an equation of motion for \(Q\),
2. find, in terms of \(h\) only, the height above the ground at which \(P\) first comes to instantaneous rest.
3. State one limitation of modelling the balls as particles that could affect your answer to part (b). In reality, the string will not be inextensible.
4. State how this would affect the accelerations of the particles.

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2. \begin{figure}[h] \end{figure} A particle \(P\) has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude \(F\) newtons, as shown in Figure 1.

1. Find the magnitude of the normal reaction of the plane on \(P\) The particle is accelerating along the plane at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(F\) The coefficient of friction between \(P\) and the plane is \(\mu\)
3. Find the value of \(\mu\), giving your answer to 2 significant figures.