Forces in vector form (i, j notation)

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 \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

12 A particle \(P\) lies at rest on a smooth horizontal table. A constant resultant force, \(\mathbf { F }\) newtons, is then applied to P .
As a result \(P\) moves in a straight line with constant acceleration \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 2 }\)
12

1. Show that the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\) 12
2. Find the speed of \(P\) after 3 seconds.
   12
3. Given that \(\mathbf { F } = \left[ \begin{array} { c } 2 \\ 1.5 \end{array} \right] \mathrm { N }\), find the mass of P .

13 A resultant force of \(\left[ \begin{array} { c } - 2 \\ 6 \end{array} \right] \mathrm { N }\) acts on a particle.
The acceleration of the particle is \(\left[ \begin{array} { c } - 6 \\ y \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }\)
Find the value of \(y\)
Circle your answer.
[0pt] [1 mark] 231018

12 A particle, under the action of two constant forces, is moving across a perfectly smooth horizontal surface at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The first force acting on the particle is \(( 400 \mathbf { i } + 180 \mathbf { j } ) \mathbf { N }\).
The second force acting on the particle is \(( p \mathbf { i } - 180 \mathbf { j } ) \mathrm { N }\).
Find the value of \(p\).
Circle your answer. \(- 400 - 390390400\)

15 Two forces, \(\mathbf { F } _ { \mathbf { 1 } }\) and \(\mathbf { F } _ { \mathbf { 2 } }\), are acting on a particle of mass 3 kilograms. It is given that $$\mathbf { F } _ { \mathbf { 1 } } = \left[ \begin{array} { c } a \\ 23 \end{array} \right] \text { newtons and } \mathbf { F } _ { \mathbf { 2 } } = \left[ \begin{array} { l } 4 \\ b \end{array} \right] \text { newtons }$$ where \(a\) and \(b\) are constants. The particle has an acceleration of \(\left[ \begin{array} { c } 4 b \\ a \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }\) Find the value of \(a\) and the value of \(b\)

10. Two forces \(\mathbf { F }\) and \(\mathbf { G }\) acting on an object are such that $$\begin{aligned} & \mathbf { F } = \mathbf { i } - 8 \mathbf { j } \\ & \mathbf { G } = 3 \mathbf { i } + 11 \mathbf { j } \end{aligned}$$ The object has a mass of 3 kg . Calculate the magnitude and direction of the acceleration of the object.