3.03d Newton's second law: 2D vectors

3 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 \(\binom { 2 } { 3 } \mathrm {~ms} ^ { - 2 }\).

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

5 A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N . \begin{figure}[h] \end{figure}

1. Find the upward acceleration.
2. Find the reaction on the man of the lift floor.

3 The three forces \(\left( \begin{array} { r } - 1 \\ 14 \\ - 8 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 3 \\ - 9 \\ 10 \end{array} \right) \mathrm { N }\) and \(\mathbf { F } \mathrm { N }\) act on a body of mass 4 kg in deep space and give it an acceleration of \(\left( \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 2 }\).

1. Calculate \(\mathbf { F }\). At one instant the velocity of the body is \(\left( \begin{array} { r } - 3 \\ 3 \\ 6 \end{array} \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Calculate the velocity and also the speed of the body 3 seconds later.

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.

8 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]{6cca1e5e-82b0-487d-8048-b9db7745dea6-5_502_935_1411_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \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 .

   4	□ box P □
   \multirow[t]{10}{*}{4 }
   4

3 Force \(\mathbf { F }\) is \(\left( \begin{array} { r } - 2 \\ 3 \\ - 4 \end{array} \right) \mathrm { N }\), force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6 \\ y \\ z \end{array} \right) \mathrm { N }\) and force \(\mathbf { H }\) is \(\left( \begin{array} { r } 3 \\ - 5 \\ - 1 \end{array} \right) \mathrm { N }\).

1. Given that \(\mathbf { F }\) and \(\mathbf { G }\) act in parallel lines, find \(y\) and \(z\). Forces \(\mathbf { F }\) and \(\mathbf { H }\) are the only forces acting on an object of mass 5 kg .
2. Calculate the acceleration of the object. Calculate also the magnitude of this acceleration.

8 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
   The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_305_1191_1050_701} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_355_1294_2156_429} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the trolleys, stating whether this rod is in tension or thrust (compression).

3 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are east, north and vertically upwards.
Forces \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) are given by \(\mathbf { p } = \left( \begin{array} { r } - 1 \\ - 1 \\ 5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1 \\ - 4 \\ 2 \end{array} \right) \mathrm { N }\) and \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 5 \\ 0 \end{array} \right) \mathrm { N }\).

1. Find which of \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) has the greatest magnitude.
2. A particle has mass 0.4 kg . The forces acting on it are \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and its weight. Find the magnitude of the particle's acceleration and describe the direction of this acceleration.

1 A uniform disc with centre \(O\) has mass \(m\) and radius \(a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(O\). One end of a light inextensible string is attached to a point on the circumference and is wrapped several times round the circumference. A particle \(P\), of mass \(2 m\), is attached to the free end of the string and the disc is held at rest with \(P\) hanging freely. The system is released from rest. Assuming that resistances may be neglected, find the acceleration of \(P\).

3 A particle \(P\), of mass \(m\), is placed at the highest point of a fixed solid smooth sphere with centre \(O\) and radius \(a\). The particle \(P\) is given a horizontal speed \(u\) and it moves in part of a vertical circle, with centre \(O\), on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the surface of the sphere, the speed of \(P\) is \(v\) and the reaction of the sphere on \(P\) has magnitude \(R\). Show that \(R = m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the sphere at the instant when \(v = 2 u\). Find \(u\) in terms of \(a\) and \(g\).

2 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-2_309_520_941_744} A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes \(2.5 \mathrm {~N} , F \mathrm {~N}\) and 2.4 N act on the particle on bearings \(\theta ^ { \circ } , 180 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram). The particle is in equilibrium.

1. Find \(\theta\) and \(F\). The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg .
2. Find the magnitude and direction of the acceleration of the particle.

5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of \(\theta\) to the horizontal where \(\sin \theta = 0.05\). The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.

1. It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
   On another occasion, the van and trailer ascend a straight hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.09\). The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances to motion are unchanged.
2. 1. Find the acceleration of the van and the tension in the tow-bar.
   2. Find the speed of the van when it has travelled a distance of 375 m up the hill.

4 A skier of mass 80 kg is pulled up a slope which makes an angle of \(20 ^ { \circ }\) with the horizontal. The skier is subject to a constant frictional force of magnitude 70 N . The speed of the skier increases from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(A\) to \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\), and the distance \(A B\) is 25 m .

1. By modelling the skier as a small object, calculate the work done by the pulling force as the skier moves from \(A\) to \(B\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-2_451_1019_1425_603} It is given that the pulling force has constant magnitude \(P \mathrm {~N}\), and that it acts at a constant angle of \(30 ^ { \circ }\) above the slope (see diagram). Calculate \(P\).

5 A model train has mass 100 kg . When the train is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the resistance to its motion is \(3 v ^ { 2 } \mathrm {~N}\) and the power output of the train is \(\frac { 3000 } { v } \mathrm {~W}\).

1. Show that the driving force acting on the train is 120 N at an instant when the train is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the acceleration of the train at an instant when it is moving horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train moves with constant speed up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 98 }\).
3. Calculate the speed of the train.

7 \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-4_782_1006_274_571} One end of a light inextensible string of length 0.8 m is attached to a fixed point \(A\) which lies above a smooth horizontal table. The other end of the string is attached to a particle \(P\), of mass 0.3 kg , which moves in a horizontal circle on the table with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 } . A P\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram).

1. Calculate the tension in the string.
2. Calculate the normal contact force between the particle and the table. The particle now moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is on the point of leaving the surface of the table.
3. Calculate \(v\).

4 A car of mass 1200 kg has a maximum speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when travelling on a horizontal road. The car experiences a resistance of \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The maximum power of the car's engine is 45000 W .

1. Show that \(k = 50\).
2. Find the maximum possible acceleration of the car when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal road.
3. The car climbs a hill, which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal, at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the power of the car's engine.

4 A car of mass 800 kg experiences a resistance of magnitude \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed. The car's engine is working at a constant rate of \(P \mathrm {~W}\). At an instant when the car is travelling on a horizontal road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At an instant when the car is ascending a hill of constant slope \(12 ^ { \circ }\) to the horizontal with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(k = 0.900\), correct to 3 decimal places, and find \(P\). The power is increased to \(1.5 P \mathrm {~W}\).
2. Calculate the maximum steady speed of the car on a horizontal road.

5 \includegraphics[max width=\textwidth, alt={}, center]{dd23f4a8-f7e7-4f80-bad6-7e9ec21565fc-3_729_739_868_703} A particle \(P\) of mass 0.2 kg is attached to one end of each of two light inextensible strings, one of length 0.4 m and one of length 0.3 m . The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The particle moves in a horizontal circle of radius 0.24 m at a constant angular speed of \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Both strings are taut, the tension in \(A P\) is \(S \mathrm {~N}\) and the tension in \(B P\) is \(T \mathrm {~N}\).

1. By resolving vertically, show that \(4 S = 3 T + 9.8\).
2. Find another equation connecting \(S\) and \(T\) and hence calculate the tensions, correct to 1 decimal place. \section*{[Questions 6 and 7 are printed overleaf.]}

4 A car of mass 700 kg is moving along a horizontal road against a constant resistance to motion of 400 N . At an instant when the car is travelling at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the driving force of the car at this instant.
2. Find the power at this instant. The maximum steady speed of the car on a horizontal road is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the maximum power of the car. The car now moves at maximum power against the same resistance up a slope of constant angle \(\theta ^ { \circ }\) to the horizontal. The maximum steady speed up the slope is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Find \(\theta\).

7 \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.2 kg is moving on the smooth inner surface of a fixed hollow hemisphere which has centre \(O\) and radius \(5 \mathrm {~m} . P\) moves with constant angular speed \(\omega\) in a horizontal circle at a vertical distance of 3 m below the level of \(O\) (see Fig.1).

1. Calculate the magnitude of the force exerted by the hemisphere on \(P\).
2. Calculate \(\omega\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_592_773_1231_687} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string is now attached to \(P\). The string passes through a small smooth hole at the lowest point of the hemisphere and a particle of mass 0.1 kg hangs in equilibrium at the end of the string. \(P\) moves in the same horizontal circle as before (see Fig. 2).
3. Calculate the new angular speed of \(P\).

2 The resistance to the motion of a car is \(k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The power exerted by the car's engine is 15000 W , and the car has constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal road.

1. Show that \(k = 4.8\). With the engine operating at a much lower power, the car descends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). At an instant when the speed of the car is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the mass of the car is 700 kg , calculate the power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_579_447_1658_849} A particle \(P\) of mass 0.4 kg is attached to one end of each of two light inextensible strings which are both taut. The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.5 m long. The string \(B P\) makes an angle of \(60 ^ { \circ }\) with the vertical. \(P\) moves with constant angular speed in a horizontal circle with centre vertically below \(B\) (see diagram). The tension in the string \(A P\) is twice the tension in the string \(B P\). Calculate

6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5 ^ { \circ }\) to the horizontal. At a certain point \(P\) on the hill the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
2. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW , correct to 3 significant figures.
3. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\).

4 A car of mass 900 kg is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a level road. The total resistance to motion is 450 N .

1. Calculate the power output of the car's engine. A roof box of mass 50 kg is mounted on the roof of the car. The total resistance to motion of the vehicle increases to 500 N .
2. The car's engine continues to work at the same rate. Calculate the maximum speed of the car on the level road. The power output of the car's engine increases to 15000 W . The resistance to motion of the car, with roof box, remains 500 N .
3. Calculate the instantaneous acceleration of the car on the level road when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. The car climbs a hill which is at an angle of \(5 ^ { \circ }\) to the horizontal. Calculate the instantaneous retardation of the car when its speed is \(26 \mathrm {~ms} ^ { - 1 }\).

8 In this question \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) denote unit vectors which are horizontal and vertically upwards respectively.
A particle of mass 5 kg , initially at rest at the point with position vector \(\binom { 2 } { 45 } \mathrm {~m}\), is acted on by gravity and also by two forces \(\binom { 15 } { - 8 } \mathrm {~N}\) and \(\binom { - 7 } { - 2 } \mathrm {~N}\).

1. Find the acceleration vector of the particle.
2. Find the position vector of the particle after 10 seconds.

2. \begin{figure}[h] \end{figure} A small ball, \(P\), of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, \(Q\), of mass 0.6 kg , that hangs freely below the pulley. Ball \(P\) is released from rest, with the rope taut, with \(P\) at a distance of 1.5 m from the pulley and with \(Q\) at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball \(Q\) descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,

1. show that the acceleration of \(Q\), as it falls, is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. find the time taken by \(P\) to hit the pulley from the instant when \(P\) is released.
3. State one limitation of the model that will affect the accuracy of your answer to part (a).