3.03p Resultant forces: using vectors

1 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_305_295_264_868} Two perpendicular forces of magnitudes \(F \mathrm {~N}\) and 8 N act at a point \(O\) (see diagram). Their resultant has magnitude 17 N .

1. Calculate \(F\) and find the angle which the resultant makes with the 8 N force. A third force of magnitude \(E \mathrm {~N}\), acting in the same plane as the two original forces, is now applied at the point \(O\). The three forces of magnitudes \(E N , F N\) and \(8 N\) are in equilibrium.
2. State the value of \(E\) and the angle between the directions of the \(E \mathrm {~N}\) and 8 N forces.

3 Two forces of magnitudes 8 N and 12 N act at a point \(O\).

1. Given that the two forces are perpendicular to each other, find
   1. the angle between the resultant and the 12 N force,
   2. the magnitude of the resultant.
   3. It is given instead that the resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts in a direction perpendicular to the 8 N force (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-2_248_388_1877_826}
      (a) Calculate the angle between the resultant and the 12 N force.
      (b) Find \(R\).

4 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).

1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
2. the magnitude of the force exerted on \(P\) by the surface,
3. the angle between the surface and the 10 N force.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendi cular unit vectors in a horizontal plane]

A particle P is moving on a smooth horizontal surface under the action of two forces.
Given that

• the mass of P is 2 kg
• the two forces are \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\), where C is a constant
• the magnitude of the acceleration of P is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) find the two possible values of C .

7 A rectangular book ABCD rests on a smooth horizontal table. The length of AB is 28 cm and the length of AD is 18 cm . The following five forces act on the book, as shown in the diagram.

• 4 N at A in the direction AD
• 5 N at B in the direction BC
• 3 N at B in the direction BA
• 9 N at D in the direction DA
• 3 N at D in the direction DC \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-06_663_830_774_242}
  1. Show that the resultant of the forces acting on the book has zero magnitude.
  2. Find the total moment of the forces about the centre of the book. Give your answer in Nm .
  3. Describe how the book will move under the action of these forces.

2 The diagram shows three forces and the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\), which all lie in the same plane. \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_415_398_1507_605} \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_172_166_1567_1217}

1. Express the resultant of the three forces in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of the resultant force.
3. Draw a diagram to show the direction of the resultant force, and find the angle that it makes with the unit vector \(\mathbf { i }\).

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}

3. A particle is in equilibrium under the action of three forces \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) acting in the same horizontal plane. \(P\) has magnitude 9 N and acts on a bearing of \(030 ^ { \circ } . Q\) has magnitude 12 N . and acts on a bearing of \(225 ^ { \circ }\).

1. Find the values of \(a\) and \(b\) such that \(\mathbf { R } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively.
2. Calculate the magnitude and direction of \(\mathbf { R }\)

2. A particle of mass 8 kg moves in a horizontal plane and is acted upon by three forces \(\mathbf { F } _ { 1 } = ( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.

1. Find the magnitude, in newtons, of the resultant force which acts on the particle, giving your answer in the form \(k \sqrt { } 5\).
2. Calculate, giving your answer in degrees correct to 1 decimal place, the angle the acceleration of the particle makes with the vector \(\mathbf { i }\).

9 Two forces, of magnitudes 2 N and 5 N , act on a particle in the directions shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-07_323_755_548_283}

1. Calculate the magnitude of the resultant force on the particle.
2. Calculate the angle between this resultant force and the force of magnitude 5 N .

7 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-3_343_401_1439_872} The diagram shows two forces of magnitudes 10 N and 15 N acting in a horizontal plane on a particle \(P\).

1. Find the component of the 15 N force which is parallel to the 10 N force.
2. Write down the component of the 15 N force which is perpendicular to the 10 N force.
3. Hence, or otherwise, calculate the magnitude and direction of the resultant force on \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-4_572_672_456_701} The diagram shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting at the origin \(O\) of rectangular coordinates \(O x y\). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 5 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, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis.

Three coplanar forces of magnitudes \(100\text{ N}\), \(50\text{ N}\) and \(50\text{ N}\) act at a point \(A\), as shown in the diagram. The value of \(\cos \alpha\) is \(\frac{4}{5}\). \includegraphics{figure_1} Find the magnitude of the resultant of the three forces and state its direction. [3]

\includegraphics{figure_6} Three coplanar forces of magnitudes 10 N, 25 N and 20 N act at a point \(O\) in the directions shown in the diagram.

1. Given that the component of the resultant force in the \(x\)-direction is zero, find \(\alpha\), and hence find the magnitude of the resultant force. [4]
2. Given instead that \(\alpha = 45\), find the magnitude and direction of the resultant of the three forces. [5]

\includegraphics{figure_2} Coplanar forces of magnitudes \(60\text{N}\), \(20\text{N}\), \(16\text{N}\) and \(14\text{N}\) act at a point in the directions shown in the diagram. Find the magnitude and direction of the resultant force. [6]

\includegraphics{figure_5} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(10\) N, \(50\) N and \(40\) N. The directions of the forces are as shown in the diagram.

1. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]
2. Given instead that \(F = 10\sqrt{2}\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force. [3]

\includegraphics{figure_2} Three coplanar forces act at a point. The magnitudes of the forces are \(5 \text{ N}\), \(6 \text{ N}\) and \(7 \text{ N}\), and the directions in which the forces act are shown in the diagram. Find the magnitude and direction of the resultant of the three forces. [6]

\includegraphics{figure_3} Forces of magnitudes 7 N, 10 N and 15 N act on a particle in the directions shown in the diagram.

1. Find the component of the resultant of the three forces
   1. in the \(x\)-direction,
   2. in the \(y\)-direction.
   [3]
2. Hence find the direction of the resultant. [2]

\includegraphics{figure_4} Coplanar forces of magnitudes 250 N, 160 N and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]

\includegraphics{figure_4} Coplanar forces of magnitudes \(250 \text{ N}\), \(160 \text{ N}\) and \(370 \text{ N}\) act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]

\includegraphics{figure_1} Four coplanar forces of magnitudes 4 N, 8 N, 12 N and 16 N act at a point. The directions in which the forces act are shown in Fig. 1.

1. Find the magnitude and direction of the resultant of the four forces. [5]

\includegraphics{figure_2} The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N and 12 N also exchange their directions (see Fig. 2).

1. State the magnitude and direction of the resultant of the four forces in Fig. 2. [2]

\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 N and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R\) N in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\). [5]
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]

\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]

Two forces, \(\mathbf{F}_1\) and \(\mathbf{F}_2\), act on a particle \(A\). \(\mathbf{F}_1 = (2i - 3j)\) N and \(\mathbf{F}_2 = (pi + qj)\) N, where \(p\) and \(q\) are constants. Given that the resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is parallel to \((\mathbf{i} + 2\mathbf{j})\),

1. show that \(2p - q + 7 = 0\) [5] Given that \(q = 11\) and that the mass of \(A\) is 2 kg, and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only forces acting on \(A\),
2. find the magnitude of the acceleration of \(A\). [5]

Two forces \(\mathbf{F_1}\) and \(\mathbf{F_2}\) act on a particle. The force \(\mathbf{F_1}\) has magnitude 8 N and acts due east. The resultant of \(\mathbf{F_1}\) and \(\mathbf{F_2}\) is a force of magnitude 14 N acting in a direction whose bearing is \(120°\). Find

1. the magnitude of \(\mathbf{F_2}\), [4]
2. the direction of \(\mathbf{F_2}\), giving your answer as a bearing to the nearest degree. [5]