3.03c Newton's second law: F=ma one dimension

3 A box, of mass 3 kg , is placed on a rough slope inclined at an angle of \(40 ^ { \circ }\) to the horizontal. It is released from rest and slides down the slope.

1. Draw a diagram to show the forces acting on the box.
2. Find the magnitude of the normal reaction force acting on the box.
3. The coefficient of friction between the box and the slope is 0.2 . Find the magnitude of the friction force acting on the box.
4. Find the acceleration of the box.
5. State an assumption that you have made about the forces acting on the box.

4 A tractor, of mass 3500 kg , is used to tow a trailer, of mass 2400 kg , across a horizontal field. The trailer is connected to the tractor by a horizontal tow bar. As they move, a constant resistance force of 800 newtons acts on the trailer and a constant resistance force of \(R\) newtons acts on the tractor. A forward driving force of 2500 newtons acts on the tractor. The trailer and tractor accelerate at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. \(\quad\) Find \(R\).
2. Find the magnitude of the force that the tow bar exerts on the trailer.
3. State the magnitude of the force that the tow bar exerts on the tractor.

2 A train travels along a straight horizontal track between two points \(A\) and \(B\).
Initially the train is at \(A\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Due to a problem, the train has to slow down and stop. At time \(t = 40\) seconds it begins to move again. At time \(t = 120\) seconds the train is at \(B\) and moving at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) again. The graph below shows how the velocity of the train varies as it moves from \(A\) to \(B\). \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-2_408_1086_1505_434}

1. Use the graph to find the total distance between the points \(A\) and \(B\).
2. The train should have travelled between \(A\) and \(B\) at a constant velocity of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Calculate the time that the train would take to travel between \(A\) and \(B\) at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   2. Calculate the time by which the train was delayed.
3. The train has mass 500 tonnes. Find the resultant force acting on the train when \(40 < t < 120\).
   (4 marks)

4 Two particles, \(A\) of mass 5 kg and \(B\) of mass 9 kg , are attached to the ends of a light inextensible string. The string passes over a light smooth pulley as shown in the diagram. The particles are released from rest at the same height. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_378_287_1580_872}

1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. When \(B\) has been moving for 0.5 seconds it hits the floor. Find the height of \(A\), above the floor, at this time. Assume that \(A\) is still below the pulley when \(B\) hits the floor.
   (4 marks)

5 A sphere of mass 200 grams is released from rest and allowed to fall vertically.

1. A student states that the acceleration of the sphere is \(9.8 \mathrm {~ms} ^ { - 2 }\) while it is falling. What modelling assumption is this student making?
2. The student conducts an experiment and finds that the acceleration of the ball is in fact \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). He formulates a model for the motion that assumes a constant resistance force acts on the ball as it is falling.
   1. Calculate the magnitude of this resistance force based on this assumption.
   2. Describe how the resistance force would vary in reality.
3. In a revised model the resistance force is assumed to be proportional to the speed of the sphere.
   1. State the initial acceleration of the sphere.
   2. State what would happen to the acceleration of the sphere if it were able to fall for a long period of time.

1 The diagram shows a velocity-time graph for a lift. \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_337_917_552_557}

1. Find the distance travelled by the lift.
2. Find the acceleration of the lift during the first 4 seconds of the motion.
3. The lift is raised by a single vertical cable. The mass of the lift is 400 kg . Find the tension in the cable during the first 4 seconds of the motion.

3 A car is travelling in a straight line on a horizontal road. A driving force, of magnitude 3000 N , acts in the direction of motion and a resistance force, of magnitude 600 N , opposes the motion of the car. Assume that no other horizontal forces act on the car.

1. Find the magnitude of the resultant force on the car.
2. The mass of the car is 1200 kg . Find the acceleration of the car. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_38_118_440_159} \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_40_118_529_159}

2 A block, of mass 10 kg , is at rest on a rough horizontal surface, when a horizontal force, of magnitude \(P\) newtons, is applied to the block, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-04_108_962_461_539} The coefficient of friction between the block and the surface is 0.5 .

1. Draw and label a diagram to show all the forces acting on the block.
2. 1. Calculate the magnitude of the normal reaction force acting on the block.
   2. Find the maximum possible magnitude of the friction force between the block and the surface.
   3. Given that \(P = 30\), state the magnitude of the friction force acting on the block.
3. Given that \(P = 80\), find the acceleration of the block.
   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-05_2484_1709_223_153}

7 A particle, of mass 10 kg , moves on a smooth horizontal surface. A single horizontal force, \(( 9 \mathbf { i } + 12 \mathbf { j } )\) newtons, acts on the particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the acceleration of the particle.
2. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the velocity of the particle is \(( 2.2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the particle is at the origin.
   1. Find the distance between the particle and the origin when \(t = 5\).
   2. Express \(\mathbf { v }\) in terms of \(t\).
   3. Find \(t\) when the particle is travelling north-east.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-15_2484_1709_223_153}

1 A crane is used to lift a load, using a single vertical cable which is attached to the load. The load accelerates uniformly from rest. When it has risen 0.9 metres, its speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. 1. Show that the acceleration of the load is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the time taken for the load to rise 0.9 metres.
2. Given that the mass of the load is 800 kg , find the tension in the cable while the load is accelerating.

2 A wooden block, of mass 4 kg , is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.3 . A horizontal force, of magnitude 30 newtons, acts on the block and causes it to accelerate. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-2_111_771_1146_639}

1. Draw a diagram to show all the forces acting on the block.
2. Calculate the magnitude of the normal reaction force acting on the block.
3. Find the magnitude of the friction force acting on the block.
4. Find the acceleration of the block.

8 Three forces act in a vertical plane on an object of mass 250 kg , as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-5_481_1139_408_447} The two forces \(P\) newtons and \(Q\) newtons each act at \(80 ^ { \circ }\) to the horizontal. The object accelerates horizontally at \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) under the action of these forces.

1. Show that $$P = 125 \left( \frac { a } { \cos 80 ^ { \circ } } + \frac { g } { \sin 80 ^ { \circ } } \right)$$
2. Find the value of \(a\) for which \(Q\) is zero.

3 A car is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The driver applies the brakes and a constant braking force acts on the car until it comes to rest.

1. Assume that no other horizontal forces act on the car.
   1. After the car has travelled 75 metres, its speed has reduced to \(10 \mathrm {~ms} ^ { - 1 }\). Find the acceleration of the car.
   2. Find the time taken for the speed of the car to reduce from \(20 \mathrm {~ms} ^ { - 1 }\) to zero.
   3. Given that the mass of the car is 1400 kg , find the magnitude of the constant braking force.
2. Given that a constant air resistance force of magnitude 200 N acts on the car during the motion, find the magnitude of the constant braking force.
   (1 mark)

1 A car is travelling along a straight horizontal road. It is moving at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it starts to accelerate. It accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 12 seconds.

1. Find the speed of the car at the end of the 12 seconds.
2. Find the distance travelled during the 12 seconds.
3. The mass of the car is 1400 kg . A horizontal forward driving force of 1600 N acts on the car during the 12 seconds. Find the magnitude of the resistance force that acts on the car.
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-02_1513_1709_1192_153}

1 A car of mass 1000 kg is travelling along a straight, level road. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the car when a resultant force of 2000 N acts on it in the direction of its motion. How long does it take the car to increase its speed from \(5 \mathrm {~ms} ^ { - 1 }\) to \(12.5 \mathrm {~ms} ^ { - 1 }\) ? The car has an acceleration of \(1.4 \mathrm {~ms} ^ { - 2 }\) when there is a driving force of 2000 N .
2. Show that the resistance to motion of the car is 600 N . A trailer is now atached to the car, as shown in Fig. 6.2. The car still has a driving force of 2000 N and resistance to motion of 600 N . The trailer has a mass of 800 kg . The tow-bar connecting the car and the trailer is light and horizontal. The car and trailer are accelerating at \(0.7 \mathrm {~ms} ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5a09ed4-a32f-4ff7-aa08-6e54c2ab26a0-1_165_883_1279_554} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Show that the resistance to the motion of the trailer is 140 N .
4. Calculate the force in the tow bar. The driving force is now removed and a braking force of 610 N is applied to the car. All the resistances to motion remain as before. The trailer has no brakes.
5. Calculate the new acceleration. Calculate also the force in the tow-bar, stating whether it is a tension or a thrust (compression).

2 Fig. 3 shows two people, Sam and Tom, pushing a car of mass 1000 kg along a straight line \(l\) on level ground. Sam pushes with a constant horizontal force of 300 N at an angle of \(30 ^ { \circ }\) to the line \(l\).
Tom pushes with a constant horizontal force of 175 N at an angle of \(15 ^ { \circ }\) to the line \(l\). \begin{figure}[h] \end{figure}

1. The car starts at rest and moves with constant acceleration. After 6 seconds it has travelled 7.2 m . Find its acceleration.
2. Find the resistance force acting on the car along the line \(l\).
3. The resultant of the forces exerted by Sam and Tom is not in the direction of the car's acceleration. Explain briefly why.

4 As shown in Fig. 4, boxes P and Q are descending vertically supported by a parachute. Box P has mass 75 kg . Box Q has mass 25 kg and hangs from box P by means of a light vertical wire. Air resistance on the boxes should be neglected. At one stage the boxes are slowing in their descent with the parachute exerting an upward vertical force of 1030 N on box P . The acceleration of the boxes is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards and the tension in the wire is \(T \mathrm {~N}\). \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on box P and another diagram showing all the forces acting on box Q .
2. Write down separate equations of motion for box P and for box Q .
3. Calculate the tension in the wire.

5 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-4_497_927_1350_646} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
5. Find the velocity of the tub when it has moved a further 1.65 m .

1 Fig. 1.1 shows a circular cylinder of mass 100 kg being raised by a light, inextensible vertical wire AB . There is negligible air resistance. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the cylinder when the tension in the wire is 1000 N .
2. Calculate the tension in the wire when the cylinder has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The cylinder is now raised inside a fixed smooth vertical tube that prevents horizontal motion but provides negligible resistance to the upward motion of the cylinder. When the wire is inclined at \(30 ^ { \circ }\) to the vertical, as shown in Fig. 1.2, the cylinder again has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_308_490_1230_849} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
3. Calculate the new tension in the wire.

2 Boxes A and B slide on a smooth, horizontal plane. Box A has a mass of 4 kg and box B a mass of 5 kg . They are connected by a light, inextensible, horizontal wire. Horizontal forces of 9 N and 135 N act on A and B in the directions shown in Fig. 5. \begin{figure}[h] \end{figure} Calculate the tension in the wire joining the boxes.

4 Two trucks, A and B, each of mass 10000 kg , are pulled along a straight, horizontal track by a constant, horizontal force of \(P \mathrm {~N}\). The coupling between the trucks is light and horizontal. This situation and the resistances to motion of the trucks are shown in Fig. 4. \begin{figure}[h] \end{figure} The acceleration of the system is \(0.2 \mathrm {~ms} ^ { 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck A is now subjected to an extra resistive force of 2000 N while \(P\) does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

5 A train consists of an engine of mass 10000 kg pulling one truck of mass 4000 kg . The coupling between the engine and the truck is light and parallel to the track. The train is accelerating at \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\) along a straight, level track.

1. What is the resultant force on the train in the direction of its motion? The driving force of the engine is 4000 N .
2. What is the resistance to the motion of the train?
3. If the tension in the coupling is 1150 N , what is the resistance to the motion of the truck? With the same overall resistance to motion, the train now climbs a uniform slope inclined at \(3 ^ { \circ }\) to the horizontal with the same acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\).
4. What extra driving force is being applied?

6 Force \(\mathbf { F } _ { 1 }\) is \(\binom { 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { 3 } { 5 }\), where \(\left. \int _ { 0 } \right] _ { \text {and } } \binom { 0 } { 1 }\) are vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.

5 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate \(\mathbf { F }\).
2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).

6 The force acting on a particle of mass 1.5 kg is given by the vector \(\binom { 6 } { 9 } \mathrm {~N}\).

1. Give the acceleration of the particle as a vector.
2. Calculate the angle that the acceleration makes with the direction \(\binom { 1 } { 0 }\).
3. At a certain point of its motion, the particle has a velocity of \(\binom { - 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the displacement of the particle over the next two seconds.