Newton's second law with vector forces (find acceleration or force)

3. A particle \(P\) of mass 0.4 kg moves under the action of a single constant force \(\mathbf { F }\) newtons. The acceleration of \(P\) is \(( 6 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the angle between the acceleration and \(\mathbf { i }\),
2. the magnitude of \(\mathbf { F }\). At time \(t\) seconds the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when \(t = 0 , \mathbf { v } = 9 \mathbf { i } - 10 \mathbf { j }\), (c) find the velocity of \(P\) when \(t = 5\).

2 A particle of mass 5 kg has constant acceleration. Initially, the particle is at \(\binom { - 1 } { 2 } \mathrm {~m}\) with velocity \(\binom { 2 } { - 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\); after 4 seconds the particle has velocity \(\binom { 12 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the acceleration of the particle.
2. Calculate the position of the particle at the end of the 4 seconds.
3. Calculate the force acting on the particle.

3 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
2. Find \(\mathbf { G }\) and \(\mathbf { H }\).

2 Force \(\mathbf { F } _ { 1 }\) is \(\binom { - 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { - 3 } { 5 } \mathrm {~N}\), where \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.

2. \begin{figure}[h] \end{figure} Figure 1 shows a regular hexagon \(O P Q R S T\).
The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O P }\) and \(\mathbf { q } = \overrightarrow { O Q }\) Forces, in Newtons, \(\mathbf { F } _ { P } = ( \overrightarrow { O P } ) , \mathbf { F } _ { Q } = 2 \times ( \overrightarrow { O Q } ) , \mathbf { F } _ { R } = 3 \times ( \overrightarrow { O R } ) , \mathbf { F } _ { S } = 4 \times ( \overrightarrow { O S } )\) and \(\mathbf { F } _ { T } = 5 \times ( \overrightarrow { O T } )\) are applied to a particle.

1. Find, in terms of \(\mathbf { p }\) and \(\mathbf { q }\), the resultant force on the particle. The magnitude of the acceleration of the particle due to these forces is \(13 \mathrm {~ms} ^ { - 2 }\) Given that the mass of the particle is 3 kg ,
2. find \(| \mathbf { p } |\) \includegraphics[max width=\textwidth, alt={}, center]{71cd126f-1c7d-4e37-a26d-7ff98a74fd79-04_2255_56_310_1980}

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

9 In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A model ship of mass 2 kg is moving so that its acceleration vector \(\mathbf { a m s } ^ { - 2 }\) at time \(t\) seconds is given by \(\mathbf { a } = 3 ( 2 t - 5 ) \mathbf { i } + 4 \mathbf { j }\). When \(t = T\), the magnitude of the horizontal force acting on the ship is 10 N . Find the possible values of \(T\).

9 Two forces \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F N }\) act on a particle \(P\) of mass 4 kg .
Given that the acceleration of \(P\) is \(( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\), calculate \(\mathbf { F }\).

1. At time \(t = 0\), a particle of mass 2 kg has velocity \(( 8 \mathbf { i } + \lambda \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors and \(\lambda > 0\).

Given that the speed of the particle at time \(t = 0\) is \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),

1. find the value of \(\lambda\). The particle experiences a constant retarding force \(\mathbf { F }\) so that when \(t = 5\), it has velocity \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Show that \(\mathbf { F }\) can be written in the form \(\mu ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) where \(\mu\) is a constant which you should find.
   (5 marks)

6 Force \(\mathbf { F } _ { 1 }\) is \(\binom { 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { 3 } { 5 }\), where \(\left. \int _ { 0 } \right] _ { \text {and } } \binom { 0 } { 1 }\) are vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.

16 A particle moves under the action of two forces, \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) It is given that $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 1.6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( k \mathbf { i } + 5 k \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(k\) is a constant.
The acceleration of the particle is \(( 3.2 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) Find \(k\) \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-25_2488_1716_219_153}

A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. When \(t = 0\), \(P\) has velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and at time \(t = 4\) s, \(P\) has velocity \((15\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\). Find

1. the acceleration of \(P\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
2. the magnitude of \(\mathbf{F}\), [4]
3. the velocity of \(P\) at time \(t = 6\) s. [3]

A particle \(P\) of mass 3 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. At \(t = 0\), \(P\) has velocity \((3\mathbf{i} - 5\mathbf{j})\) m s\(^{-1}\). At \(t = 4\) s, the velocity of \(P\) is \((-5\mathbf{i} + 11\mathbf{j})\) m s\(^{-1}\). Find

1. the acceleration of \(P\), in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [2]
2. the magnitude of \(\mathbf{F}\). [4]

At \(t = 6\) s, \(P\) is at the point \(A\) with position vector \((6\mathbf{i} - 29\mathbf{j})\) m relative to a fixed origin \(O\). At this instant the force \(\mathbf{F}\) newtons is removed and \(P\) then moves with constant velocity. Three seconds after the force has been removed, \(P\) is at the point \(B\).

1. Calculate the distance of \(B\) from \(O\). [6]

A particle of mass 5 kg has constant acceleration. Initially, the particle is at \(\begin{pmatrix} -1 \\ 2 \end{pmatrix}\) m with velocity \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\) ms\(^{-1}\); after 4 seconds the particle has velocity \(\begin{pmatrix} 12 \\ 9 \end{pmatrix}\) ms\(^{-1}\).

1. Calculate the acceleration of the particle. [2]
2. Calculate the position of the particle at the end of the 4 seconds. [3]
3. Calculate the force acting on the particle. [2]

Two particles, \(A\) and \(B\), lie at rest on a smooth horizontal plane. \(A\) has position vector \(\mathbf{r}_A = (13\mathbf{i} - 22\mathbf{j})\) metres \(B\) has position vector \(\mathbf{r}_B = (3\mathbf{i} + 2\mathbf{j})\) metres

1. Calculate the distance between \(A\) and \(B\). [2 marks]
2. Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are applied to particle \(A\), where \(\mathbf{F}_1 = (-2\mathbf{i} + 4\mathbf{j})\) newtons \(\mathbf{F}_2 = (6\mathbf{i} - 10\mathbf{j})\) newtons Given that \(A\) remains at rest, explain why \(\mathbf{F}_3 = (-4\mathbf{i} + 6\mathbf{j})\) newtons [1 mark]
3. A force of \((5\mathbf{i} - 12\mathbf{j})\) newtons, is applied to \(B\), so that \(B\) moves, from rest, in a straight line towards \(A\). \(B\) has a mass of \(0.8 \text{kg}\)
   1. Show that the acceleration of \(B\) towards \(A\) is \(16.25 \text{m s}^{-2}\) [2 marks]
   2. Hence, find the time taken for \(B\) to reach \(A\). Give your answer to two significant figures. [2 marks]

A particle of mass 2 kg moves under the action of a constant force F N, where F is given by $$\mathbf{F} = -3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the magnitude of the acceleration of the particle. [3]
2. Given that at time \(t = 0\) seconds, the position vector of the particle is \(2\mathbf{i} - 7\mathbf{j} + 9\mathbf{k}\) and it is moving with velocity \(3\mathbf{i} - 2\mathbf{j} + \mathbf{k}\), find the position vector of the particle when \(t = 2\) seconds. [3]