3.03p Resultant forces: using vectors

3. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle \(P\) are given by $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 7 \mathbf { i } - 9 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(p\) and \(q\) are constants.
Given that \(P\) is in equilibrium,

1. find the value of \(p\) and the value of \(q\). The force \(\mathbf { F } _ { 3 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Find
2. the magnitude of \(\mathbf { R }\),
3. the angle, to the nearest degree, that the direction of \(\mathbf { R }\) makes with \(\mathbf { j }\).

2. \begin{figure}[h] \end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\), act on a particle. The force \(\mathbf { P }\) has magnitude 5 N and the force \(\mathbf { Q }\) has magnitude 3 N . The angle between the directions of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(40 ^ { \circ }\), as shown in Fig. 1. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(\mathbf { F }\).

1. Find, to 3 significant figures, the magnitude of \(\mathbf { F }\).
2. Find, in degrees to 1 decimal place, the angle between the directions of \(\mathbf { F }\) and \(\mathbf { P }\).

6. \begin{figure}[h] \end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at \(O\). The angle between the lines of action of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(120 ^ { \circ }\) as shown in Figure 4. The force \(\mathbf { P }\) has magnitude 20 N and the force \(\mathbf { Q }\) has magnitude \(X\) newtons. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is the force \(\mathbf { R }\). Given that the magnitude of \(\mathbf { R }\) is \(3 X\) newtons, find, giving your answers to 3 significant figures

1. the value of \(X\),
2. the magnitude of \(( \mathbf { P } - \mathbf { Q } )\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have mass 4 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough plane, which is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 4 } { 3 }\). The coefficient of friction between \(P\) and the plane is 0.5 . The string lies along the plane and passes over a small smooth light pulley which is fixed at the top of the plane. Particle \(Q\) hangs freely at rest vertically below the pulley. The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 4. Particle \(P\) is released from rest with the string taut and slides down the plane. Given that \(Q\) has not hit the pulley, find

1. the tension in the string during the motion,
2. the magnitude of the resultant force exerted by the string on the pulley.

7. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\). The force \(\mathbf { F } _ { 1 }\) is given by \(\mathbf { F } _ { 1 } = ( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\).
Given that the resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector ( \(\mathbf { i } + 3 \mathbf { j }\) ),

1. find \(\mathbf { F } _ { 2 }\) (7) The acceleration of \(P\) is \(( 3 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( 3 \mathbf { i } - 22 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the speed of \(P\) when \(t = 3\) seconds.
   

6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively] Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\) of mass 0.5 kg . \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\).
Given that the resultant force of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is in the same direction as \(- 2 \mathbf { i } - \mathbf { j }\),

1. show that \(p - 2 q = - 16\) Given that \(q = 3\)
2. find the magnitude of the acceleration of \(P\),
3. find the direction of the acceleration of \(P\), giving your answer as a bearing to the nearest degree. XXXXXXXXXXIXITEINTIIS AREA XX女X女X女X女X DO NOT WIRIE IN THS AREA.

4. \begin{figure}[h] \end{figure} Figure 3 shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting on a particle.
The angle between the direction of \(\mathbf { P }\) and the direction of \(\mathbf { Q }\) is \(150 ^ { \circ }\) Force \(\mathbf { P }\) has magnitude \(X\) newtons.
Force \(\mathbf { Q }\) has magnitude \(5 \sqrt { 3 } \mathrm {~N}\).
The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) has magnitude \(\sqrt { 129 } \mathrm {~N}\).
Find

1. the value of \(X\).
2. the angle between \(\mathbf { Q }\) and the resultant, giving your answer to the nearest degree.

7. \begin{figure}[h] \end{figure} Two forces, \(\mathbf { P }\) and \(\mathbf { Q }\), act on a particle. The force \(\mathbf { P }\) has magnitude 8 N and the force \(\mathbf { Q }\) has magnitude 5 N . The angle between the directions of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(50 ^ { \circ }\), as shown in Figure 3. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is the force \(\mathbf { R }\).

1. Find, to 3 significant figures, the magnitude of \(\mathbf { R }\).
2. Find, to the nearest degree, the size of the angle between the direction of \(\mathbf { P }\) and the direction of \(\mathbf { R }\).

2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] Three forces, \(( - 10 \mathbf { i } + a \mathbf { j } ) \mathrm { N } , ( b \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( 2 a \mathbf { i } + 7 \mathbf { j } ) \mathrm { N }\), where \(a\) and \(b\) are constants, act on a particle \(P\) of mass 3 kg . The acceleration of \(P\) is \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)

1. Find the value of \(a\) and the value of \(b\). At time \(t = 0\) seconds the speed of \(P\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at time \(t = 4\) seconds the velocity of \(P\) is \(( 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the value of \(u\).

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 \(g\).

6. Two forces, \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\), act on a particle \(P\) of mass \(m \mathrm {~kg}\). The resultant of the two forces is \(\mathbf { R }\). Given that \(\mathbf { R }\) acts in a direction which is parallel to the vector ( \(\mathbf { i } - 2 \mathbf { j }\) ),

1. find the angle between \(\mathbf { R }\) and the vector \(\mathbf { j }\),
2. show that \(2 p + q + 3 = 0\). Given also that \(q = 1\) and that \(P\) moves with an acceleration of magnitude \(8 \sqrt { } 5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), (c) find the value of \(m\).

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.

3 \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-3_638_839_269_653} Three coplanar forces of magnitudes \(5 \mathrm {~N} , 8 \mathrm {~N}\) and 8 N act at the origin \(O\) of rectangular coordinate axes. The directions of the forces are as shown in the diagram.

1. Find the component of the resultant of the three forces in
   1. the \(x\)-direction,
   2. the \(y\)-direction.
   3. Find the magnitude and direction of the resultant.

3 Two horizontal forces \(\mathbf { X }\) and \(\mathbf { Y }\) act at a point \(O\) and are at right angles to each other. \(\mathbf { X }\) has magnitude 12 N and acts along a bearing of \(090 ^ { \circ } . \mathbf { Y }\) has magnitude 15 N and acts along a bearing of \(000 ^ { \circ }\).

1. Calculate the magnitude and bearing of the resultant of \(\mathbf { X }\) and \(\mathbf { Y }\).
2. A third force \(\mathbf { E }\) is now applied at \(O\). The three forces \(\mathbf { X } , \mathbf { Y }\) and \(\mathbf { E }\) are in equilibrium. State the magnitude of \(\mathbf { E }\), and give the bearing along which it acts.

3 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_570_679_1512_731} Three horizontal forces act at the point \(O\). One force has magnitude 7 N and acts along the positive \(x\)-axis. The second force has magnitude 9 N and acts along the positive \(y\)-axis. The third force has magnitude 5 N and acts at an angle of \(30 ^ { \circ }\) below the negative \(x\)-axis (see diagram).

1. Find the magnitudes of the components of the 5 N force along the two axes.
2. Calculate the magnitude of the resultant of the three forces. Calculate also the angle the resultant makes with the positive \(x\)-axis.

6 A particle of mass 0.04 kg is acted on by a force of magnitude \(P \mathrm {~N}\) in a direction at an angle \(\alpha\) to the upward vertical.

1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20 ^ { \circ }\) find
   1. the value of \(P\),
   2. the magnitude of the resultant,
   3. the magnitude of the acceleration of the particle.
   4. It is given instead that \(P = 0.08\) and \(\alpha = 90 ^ { \circ }\). Find the magnitude and direction of the resultant force on the particle.

2 \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-2_620_711_543_717} Forces of magnitudes 6.5 N and 2.5 N act at a point in the directions shown. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at right angles to the force of magnitude 2.5 N (see diagram).

1. Show that \(\theta = 22.6 ^ { \circ }\), correct to 3 significant figures.
2. Find the value of \(R\).

1 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_415_823_264_660} Two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) act at the origin O of rectangular coordinates Oxy (see diagram). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 14 N and 5 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 9 N and 7 N respectively.

1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis.

3 \includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-02_570_495_1114_826} Three horizontal forces of magnitudes \(12 \mathrm {~N} , 5 \mathrm {~N}\), and 9 N act along bearings \(000 ^ { \circ } , 150 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram).

1. Show that the component of the resultant of the three forces along bearing \(270 ^ { \circ }\) has magnitude 6.5 N .
2. Find the component of the resultant of the three forces along bearing \(000 ^ { \circ }\).
3. Hence find the magnitude and bearing of the resultant of the three forces.

4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at \(60 ^ { \circ }\). \begin{figure}[h] \end{figure}

1. Calculate the component of the resultant of these two forces in the direction of the 20 N force.
2. Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
3. Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.

2 Fig. 2 shows two forces acting at A. The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h] \end{figure}

1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.

2 Two horizontal forces of magnitudes 12 N and 19 N act at a point. Given that the angle between the two forces is \(60 ^ { \circ }\), calculate

1. the magnitude of the resultant force,
2. the angle between the resultant and the 12 N force.

4 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-2_325_481_1699_792} Three horizontal forces of magnitudes \(8 \mathrm {~N} , 15 \mathrm {~N}\) and 20 N act at a point. The 8 N and 15 N forces are at right angles. The 20 N force makes an angle of \(150 ^ { \circ }\) with the 8 N force and an angle of \(120 ^ { \circ }\) with the 15 N force (see diagram).

1. Calculate the components of the resultant force in the directions of the 8 N and 15 N forces.
2. Calculate the magnitude of the resultant force, and the angle it makes with the direction of the 8 N force. The directions in which the three horizontal forces act can be altered.
3. State the greatest and least possible magnitudes of the resultant force.

1 Three horizontal forces, acting at a single point, have magnitudes \(12 \mathrm {~N} , 14 \mathrm {~N}\) and 5 N and act along bearings \(000 ^ { \circ } , 090 ^ { \circ }\) and \(270 ^ { \circ }\) respectively. Find the magnitude and bearing of their resultant.

1 Two perpendicular forces have magnitudes 8 N and 15 N . Calculate the magnitude of the resultant force, and the angle which the resultant makes with the larger force.