3.03a Force: vector nature and diagrams

1 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$

1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
2. Interpret your answer to part (i) in the cases
   (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
   (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.

2 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$

1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
2. Interpret your answer to part (i) in the cases
   (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
   (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.

4. A small stone, of mass 600 grams, is released from rest a height of 2 metres above ground level and falls under gravity. The time it takes to reach the ground is \(T\) seconds. The stone is then again released from rest at the surface of a tank containing a 2 metre depth of liquid and reaches the bottom after \(2 T\) seconds. It may be assumed that the resisting force acting on the stone is constant.

1. Find the magnitude of the resisting force exerted on the stone by the liquid.
2. Find the speed with which the stone hits the bottom of the tank.

1

1. Fig. 1.1 and Fig. 1.2 show rigid rods with forces acting as marked. The diagrams are to scale, and in each figure the side length of a grid square is 1 metre. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_428_552_443_319} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_431_553_440_1005} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
   • On the copy of Fig. 1.1 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents an anti-clockwise couple of magnitude 24 Nm .
   • On the copy of Fig. 1.2 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents a clockwise couple of magnitude 1 Nm .
   • Fig. 1.3 shows a rectangular lamina with two coplanar forces acting as marked. Each grid square has side length 1 m .
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_561_761_1452_315} \captionsetup{labelformat=empty} \caption{Fig. 1.3}
   \end{figure} A third coplanar force, of magnitude \(T \mathrm {~N}\), acts at A so that the resultant force on the lamina is zero.
   1. Calculate the value of \(T\).
   2. Determine the magnitude and direction of the couple represented by this system of three forces.

2 Three forces, of magnitudes \(33 \mathrm {~N} , 45 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. The system is in equilibrium. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-3_501_703_342_239}

1. Draw a triangle of forces for the system shown above. Your diagram should include the magnitudes of the forces ( \(33 \mathrm {~N} , 45 \mathrm {~N}\) and \(P \mathrm {~N}\) ) and angle \(\theta\).
2. If \(P = 38\), find, in degrees, the value of \(\theta\).
3. If \(\theta = 40 ^ { \circ }\), determine the possible values for \(P\).

1 Two horizontal forces of magnitudes 7 N and 15 N act at a point O .
The 15 N force acts an angle of \(\theta ^ { \circ }\) above the positive \(x\)-axis.
The 7 N force acts at an angle of \(70 ^ { \circ }\) below the negative \(x\)-axis (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-2_606_773_402_239} The resultant of the two forces acts only in the positive \(x\)-direction.

1. Calculate the value of \(\theta\).
2. Calculate the magnitude of the resultant of the two forces.

3 The diagram shows a uniform beam AB , of weight 80 N and length 7 m , resting in equilibrium in a vertical plane. The end A is in contact with a rough vertical wall, and the angle between the beam and the upward vertical is \(60 ^ { \circ }\). The beam is supported by a smooth peg at a point C , where \(\mathrm { AC } = 2 \mathrm {~m}\). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-4_474_709_445_244}

1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on the beam.
2. 1. Show that the magnitude of the frictional force exerted on the beam by the wall is 25 N .
   2. Hence determine the magnitude of the total contact force exerted on the beam by the wall.
3. Determine the direction of the total contact force exerted on the beam by the wall. The coefficient of friction between the beam and the wall is \(\mu\).
4. Find the range of possible values for \(\mu\).
5. Explain how your answer to part (b)(ii) would change if the peg were situated closer to A but the angle between the beam and the upward vertical remained at \(60 ^ { \circ }\).

2 The diagram below shows the cross-section through the centre of mass of a uniform block of weight \(W \mathrm {~N}\), resting on a slope inclined at an angle \(\alpha\) to the horizontal. The cross-section is a rectangle ABCD . The slope exerts a frictional force of magnitude \(F \mathrm {~N}\) and a normal contact force of magnitude \(R \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-3_546_940_450_242}

1. Explain why a triangle of forces may be used to model the scenario.
2. In the space provided in the Printed Answer Booklet, draw such a triangle, fully annotated, including the angle \(\alpha\) in the correct position. The coefficient of friction between the block and the slope is \(\mu\).
3. Given that the block is in limiting equilibrium, use your diagram in part (b) to show that \(\mu = \tan \alpha\). It is given that \(\mathrm { AB } = 8.9 \mathrm {~cm}\) and \(\mathrm { AD } = 11.6 \mathrm {~cm}\). The coefficient of friction between the slope and the block is 1.35 . The slope is slowly tilted so that \(\alpha\) increases.
4. Determine whether the block topples first without sliding or slides first without toppling.

4 A ball is released from rest at a height of 15 metres above ground level.

1. Find the speed of the ball when it hits the ground, assuming that no air resistance acts on the ball.
2. In fact, air resistance does act on the ball. Assume that the air resistance force has a constant magnitude of 0.9 newtons. The ball has a mass of 0.5 kg .
   1. Draw a diagram to show the forces acting on the ball, including the magnitudes of the forces acting.
   2. Show that the acceleration of the ball is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   3. Find the speed at which the ball hits the ground.
   4. Explain why the assumption that the air resistance force is constant may not be valid.

5 The constant forces \(\mathbf { F } _ { 1 } = ( 8 \mathbf { i } + 12 \mathbf { j } )\) newtons and \(\mathbf { F } _ { 2 } = ( 4 \mathbf { i } - 4 \mathbf { j } )\) newtons act on a particle. No other forces act on the particle.

1. Find the resultant force acting on the particle.
2. Given that the mass of the particle is 4 kg , show that the acceleration of the particle is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
3. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. When \(t = 20 , \mathbf { v } = 40 \mathbf { i } + 32 \mathbf { j }\). Show that \(\mathbf { v } = - 20 \mathbf { i } - 8 \mathbf { j }\) when \(t = 0\).
   2. Write down an expression for \(\mathbf { v }\) at time \(t\).
   3. Find the times when the speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

6 A box, of mass 3 kg , is placed on a slope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.

1. A simple model assumes that the slope is smooth.
   1. Draw a diagram to show the forces acting on the box.
   2. Show that the acceleration of the box is \(4.9 \mathrm {~ms} ^ { - 2 }\).
2. A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
   1. Show that the acceleration of the box is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the magnitude of the friction force acting on the box.
   3. Find the coefficient of friction between the box and the slope.
   4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.

7. \begin{figure}[h] \end{figure} A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.
The handle is inclined at an angle \(\alpha\) to the floor, as shown in Figure 1, where \(\tan \alpha = \frac { 3 } { 4 }\) The tension in the handle is 40 N .
The coefficient of friction between the crate and the floor is 0.14
The crate is modelled as a particle and the handle is modelled as a light rod.
Using the model,

1. find the acceleration of the crate. The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle \(\alpha\) to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_220_923_1457_571} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
2. Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).

9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}

1. Copy and complete the diagram to show all the forces acting on the particle.
2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.

10 A small body of mass \(m\) is thrown vertically upwards with initial velocity \(u\). Resistance to motion is \(k v ^ { 2 }\) per unit mass, where the velocity is \(v\) and \(k\) is a positive constant. Find, in terms of \(u , g\) and \(k\),

1. the time taken to reach the greatest height,
2. the greatest height to which the body will rise.

6 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-4_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.

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.

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]

\includegraphics{figure_3} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m\) kg is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac{4}{3}\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac{1}{4}\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,

1. find the value of \(m\), [10]
2. find the magnitude of the force exerted on the pulley by the string, [4]
3. find the direction of the force exerted on the pulley by the string. [1]

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]

\includegraphics{figure_2} Figure 2 shows two particles \(A\) and \(B\), of masses \(3m\) and \(4m\) respectively, attached to the ends of a light inextensible string. Initially \(A\) is held at rest on the surface of a fixed rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth light pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely and is vertically below \(P\). The system is released from rest with the string taut and with \(B\) at a height of 1.75 m above the ground. In the subsequent motion, \(A\) does not hit the pulley. For the period before \(B\) hits the ground,

1. write down an equation of motion for each particle. [4]
2. Hence show that the acceleration of \(B\) is \(\frac{8}{35}g\). [5]
3. Explain how you have used the fact that the string is inextensible in your calculation. [1]

When \(B\) hits the ground, \(B\) does not rebound and comes immediately to rest.

1. Find the distance travelled by \(A\) from the instant when the system is released to the instant when \(A\) first comes to rest. [7]

A particle \(P\), of mass 3 kg, moves under the action of two constant forces (6\(\mathbf{i}\) + 2\(\mathbf{j}\)) N and (3\(\mathbf{i}\) - 5\(\mathbf{j}\)) N.

1. Find, in the form (\(a\mathbf{i}\) + \(b\mathbf{j}\)) N, the resultant force \(\mathbf{F}\) acting on \(P\). [1]
2. Find, in degrees to one decimal place, the angle between \(\mathbf{F}\) and \(\mathbf{j}\). [3]
3. Find the acceleration of \(P\), giving your answer as a vector. [2]

The initial velocity of \(P\) is (-2\(\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\).

1. Find, to 3 significant figures, the speed of \(P\) after 2 s. [5]

Two forces \(\mathbf{P}\) and \(\mathbf{Q}\) act on a particle. The force \(\mathbf{P}\) has magnitude \(7\) N and acts due north. The resultant of \(\mathbf{P}\) and \(\mathbf{Q}\) is a force of magnitude \(10\) N acting in a direction with bearing \(120°\). Find

1. the magnitude of \(\mathbf{Q}\),
2. the direction of \(\mathbf{Q}\), giving your answer as a bearing.

[9]

\includegraphics{figure_1} A particle of weight 24 N is held in equilibrium by two light inextensible strings. One string is horizontal. The other string is inclined at an angle of 30° to the horizontal, as shown in Figure 1. The tension in the horizontal string is \(Q\) newtons and the tension in the other string is \(P\) newtons. Find

1. the value of \(P\), [3]
2. the value of \(Q\). [3]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of mass \(4\) kg and \(6\) kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac{2}{5}\). A constant force of magnitude \(40\) N is then applied to \(Q\) in the direction \(PQ\), as shown in Fig. 4.

1. Show that the acceleration of \(Q\) is \(1.2\) m s\(^{-2}\). [4]
2. Calculate the tension in the string when the system is moving. [3]
3. State how you have used the information that the string is inextensible. [1]

After the particles have been moving for \(7\) s, the string breaks. The particle \(Q\) remains under the action of the force of magnitude \(40\) N.

1. Show that \(P\) continues to move for a further \(3\) seconds. [5]
2. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. [4]

\includegraphics{figure_1} A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(AC\). The bead \(B\) is vertically below \(C\) and \(\angle BAC = \alpha\), as shown in Figure 1. Given that \(\tan \alpha = \frac{3}{4}\), find

1. the tension in the string, [3]
2. the weight of the bead. [4]