Forces in vector form: resultant and acceleration

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.

1. A particle \(P\) rests in equilibrium on a smooth horizontal plane.

A system of three forces, \(\mathbf { F } _ { 1 } \mathrm {~N} , \mathbf { F } _ { 2 } \mathrm {~N}\) and \(\mathbf { F } _ { 3 } \mathrm {~N}\) where $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 3 c \mathbf { i } + 4 c \mathbf { j } ) \\ & \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } + 7 \mathbf { j } ) \end{aligned}$$ is applied to \(P\).
Given that \(P\) remains in equilibrium,

1. find \(\mathbf { F } _ { 3 }\) in terms of \(c\), \(\mathbf { i }\) and \(\mathbf { j }\). The force \(\mathbf { F } _ { 3 }\) is removed from the system.
   Given that \(c = 2\)
2. find the size of the angle between the direction of \(\mathbf { i }\) and the direction of the resultant force acting on \(P\). The mass of \(P\) is \(m \mathrm {~kg}\).
   Given that the magnitude of the acceleration of \(P\) is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
3. find the value of \(m\).

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 .

5 In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2.5 kg .

1. Write the weight of the particle as a vector. The particle moves under the action of its weight and two external forces ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) N and \(( - \mathbf { i } + 18 \mathbf { j } ) N\).
2. Find the acceleration of the particle, giving your answer in vector form.

7 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A canal narrowboat of mass 9 tonnes is pulled by two ropes. The tensions in the ropes are \(( 450 \mathbf { i } + 20 \mathbf { j } ) \mathbf { N }\) and \(( 420 \mathbf { i } - 20 \mathbf { j } ) \mathbf { N }\). The boat experiences a resistance to motion \(\mathbf { R }\) of magnitude 300 N .

1. Explain what it means to model the boat as a particle. The boat is travelling in a straight line due east.
2. Find the equation of motion of the boat.
3. Find the acceleration of the boat giving your answer as a vector.

2 Three forces act on a particle. These forces are ( \(9 \mathbf { i } - 3 \mathbf { j }\) ) newtons, ( \(5 \mathbf { i } + 8 \mathbf { j }\) ) newtons and ( \(- 7 \mathbf { i } + 3 \mathbf { j }\) ) newtons. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the resultant of these forces.
2. Find the magnitude of the resultant force.
3. Given that the particle has mass 5 kg , find the magnitude of the acceleration of the particle.
4. Find the angle between the resultant force and the unit vector \(\mathbf { i }\).

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 \(\mathbf { i }\).

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 \(\binom { 0 } { 1 }\).

A particle of mass \(0.5\) kg moves in a straight line under the action of a variable force. At time \(t\) seconds, the force is \((3t - 2)\) N in the direction of motion. Given that the particle starts from rest, find the velocity of the particle when \(t = 4\). [5]

A particle \(P\) of mass \(0.4 \text{ kg}\) is projected horizontally along a smooth horizontal plane from a point \(O\). At time \(t \text{ s}\) after projection the velocity of \(P\) is \(v \text{ ms}^{-1}\). A force of magnitude \(0.8t \text{ N}\) directed away from \(O\) acts on \(P\) and a force of magnitude \(2e^{-t} \text{ N}\) opposes the motion of \(P\).

1. Show that \(\frac{dv}{dt} = 2t - 5e^{-t}\). [2]
2. Given that \(v = 8\) when \(t = 1\), express \(v\) in terms of \(t\). [3]
3. Find the speed of projection of \(P\). [2]

A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) above horizontal ground. At time \(t\) s after its release the velocity of \(P\) is 7.5 m s\(^{-1}\) downwards. A vertically downwards force of magnitude 0.6t N acts on \(P\). A vertically upwards force of magnitude \(ke^{-t}\) N, where \(k\) is a constant, also acts on \(P\).

1. Show that \(\frac{dv}{dt} = 10 - 5ke^{-t} + 3t\). [2]
2. Find the greatest value of \(k\) for which \(P\) does not initially move upwards. [3]
3. Given that \(k = 1\), and that \(P\) strikes the ground when \(t = 2\), find the height of \(O\) above the ground. [5]

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]

A particle \(P\) of mass \(2 \text{ kg}\) moves in a plane under the action of a single constant force \(\mathbf{F}\) newtons. At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). When \(t = 0\), \(\mathbf{v} = (-5\mathbf{i} + 7\mathbf{j})\) and when \(t = 3\), \(\mathbf{v} = (\mathbf{i} - 2\mathbf{j})\).

1. Find in degrees the angle between the direction of motion of \(P\) when \(t = 3\) and the vector \(\mathbf{j}\). [3]
2. Find the acceleration of \(P\). [2]
3. Find the magnitude of \(\mathbf{F}\). [3]
4. Find in terms of \(t\) the velocity of \(P\). [2]
5. Find the time at which \(P\) is moving parallel to the vector \(\mathbf{i} + \mathbf{j}\). [3]

A particle is acted upon by two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\), given by \(\mathbf{F}_1 = (\mathbf{i} - 3\mathbf{j})\) N, \(\mathbf{F}_2 = (p\mathbf{i} + 2p\mathbf{j})\) N, where \(p\) is a positive constant.

1. Find the angle between \(\mathbf{F}_2\) and \(\mathbf{j}\). [2]

The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\). Given that \(\mathbf{R}\) is parallel to \(\mathbf{i}\),

1. find the value of \(p\). [4]

A particle \(P\), of mass \(2.5\) kg, initially at rest at the point \(O\), moves on a smooth horizontal surface with constant acceleration \((\mathbf{i} + 2\mathbf{j})\) ms\(^{-2}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due East and due North respectively. Find

1. the velocity vector of \(P\) at time \(t\) seconds after it leaves \(O\), \hfill [2 marks]
2. the magnitude and direction of the velocity of \(P\) when \(t = 7\), \hfill [3 marks]
3. the magnitude, in N, of the force acting on \(P\). \hfill [2 marks]

Two forces \(\mathbf{F}\) and \(\mathbf{G}\) are given by \(\mathbf{F} = (6\mathbf{i} - 5\mathbf{j})\) N, \(\mathbf{G} = (3\mathbf{i} + 17\mathbf{j})\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the \(x\) and \(y\) directions respectively and the unit of length on each axis is 1 cm.

1. Find the magnitude of \(\mathbf{R}\), the resultant of \(\mathbf{F}\) and \(\mathbf{G}\). [3 marks]
2. Find the angle between the direction of \(\mathbf{R}\) and the positive \(x\)-axis. [2 marks]

\(\mathbf{R}\) acts through the point \(P(-4, 3)\). \(O\) is the origin \((0, 0)\).

1. Use the fact that \(OP\) is perpendicular to the line of action of \(\mathbf{R}\) to calculate the moment of \(\mathbf{R}\) about an axis through the origin and perpendicular to the \(x\)-\(y\) plane. [3 marks]

A constant force, \(\mathbf{F}\), acts on a particle, \(P\), of mass 5 kg causing its velocity to change from \((-2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) to \((4\mathbf{i} - 7\mathbf{j})\) m s\(^{-1}\) in 2 seconds.

1. Find, in the form \(a\mathbf{i} + b\mathbf{j}\), the acceleration of \(P\). [2 marks]
2. Show that the magnitude of \(\mathbf{F}\) is 25 N and find, to the nearest degree, the acute angle between the line of action of \(\mathbf{F}\) and the vector \(\mathbf{j}\). [5 marks]

A particle \(P\) of mass 3 kg has position vector \(\mathbf{r} = (2t^2 - 4t)\mathbf{i} + (1 - t^2)\mathbf{j}\) m at time \(t\) seconds.

1. Find the velocity vector of \(P\) when \(t = 3\). [3 marks]
2. Find the magnitude of the force acting on \(P\), showing that this force is constant. [4 marks]

A particle \(P\) of mass \(m \text{kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \((-4\mathbf{i} + 2\mathbf{j}) \text{N}\) and \((a\mathbf{i} + b\mathbf{j}) \text{N}\). The resultant of these two forces is \(\mathbf{R} \text{N}\). It is given that \(\mathbf{R}\) acts in a direction which is parallel to the vector \(-\mathbf{i} + 3\mathbf{j}\).

1. Show that \(3a + b = 10\). [3]

It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5\sqrt{10} \text{ms}^{-2}\).

1. Determine the value of \(m\). [4]

Two forces \(\begin{bmatrix}3\\-2\end{bmatrix}\) N and \(\begin{bmatrix}-7\\-5\end{bmatrix}\) N act on a particle. Find the resultant force. Circle your answer. [1 mark] \(\begin{bmatrix}-21\\10\end{bmatrix}\) N \(\begin{bmatrix}-4\\-7\end{bmatrix}\) N \(\begin{bmatrix}4\\3\end{bmatrix}\) N \(\begin{bmatrix}10\\7\end{bmatrix}\) N

Two points \(A\) and \(B\) lie in a horizontal plane and have coordinates \((-2, 7)\) and \((3, 19)\) respectively. A particle moves in a straight line from \(A\) to \(B\) under the action of a constant resultant force of magnitude 6.5 N Express the resultant force in vector form. [3 marks]

A resultant force of \(\begin{bmatrix} -2 \\ 6 \end{bmatrix}\) N acts on a particle. The acceleration of the particle is \(\begin{bmatrix} -6 \\ y \end{bmatrix} \text{ m s}^{-2}\) Find the value of \(y\) Circle your answer. [1 mark] \(2\) \quad \(3\) \quad \(10\) \quad \(18\)

It is given that two points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 5 \\ -1 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} 13 \\ 5 \end{bmatrix} \text{ metres.}$$

1. Show that the distance from \(A\) to \(B\) is 10 metres. [3 marks]
2. A constant resultant force, of magnitude \(R\) newtons, acts on a particle so that it moves in a straight line passing through the same two points \(A\) and \(B\) At \(A\), the speed of the particle is 3 m s\(^{-1}\) in the direction from \(A\) to \(B\) The particle takes 2 seconds to travel from \(A\) to \(B\) The mass of the particle is 150 grams. Find the value of \(R\) [3 marks]

A number of forces act on a particle such that the resultant force is \(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\) N One of the forces acting on the particle is \(\begin{pmatrix} 8 \\ -5 \end{pmatrix}\) N Calculate the total of the other forces acting on the particle. Circle your answer. \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\) N \quad \(\begin{pmatrix} 14 \\ -8 \end{pmatrix}\) N \quad \(\begin{pmatrix} -2 \\ 2 \end{pmatrix}\) N \quad \(\begin{pmatrix} -14 \\ 8 \end{pmatrix}\) N [1 mark]