Forces as vectors

3 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.

1. Find the values of \(x , y\) and \(z\).
2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).

2 Force \(\mathbf { F }\) is \(\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right) \mathrm { N }\) and force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6 \\ 2 \\ 4 \end{array} \right) \mathrm { N }\).

1. Find the resultant of \(\mathbf { F }\) and \(\mathbf { G }\) and calculate its magnitude.
2. Forces \(\mathbf { F }\), \(2 \mathbf { G }\) and \(\mathbf { H }\) act on a particle which is in equilibrium. Find \(\mathbf { H }\).

5 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.

1. Find the values of \(x , y\) and \(z\).
2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).

3 Force \(\mathbf { F }\) is \(\left( \begin{array} { r } - 2 \\ 3 \\ - 4 \end{array} \right) \mathrm { N }\), force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6 \\ y \\ z \end{array} \right) \mathrm { N }\) and force \(\mathbf { H }\) is \(\left( \begin{array} { r } 3 \\ - 5 \\ - 1 \end{array} \right) \mathrm { N }\).

1. Given that \(\mathbf { F }\) and \(\mathbf { G }\) act in parallel lines, find \(y\) and \(z\). Forces \(\mathbf { F }\) and \(\mathbf { H }\) are the only forces acting on an object of mass 5 kg .
2. Calculate the acceleration of the object. Calculate also the magnitude of this acceleration.

3 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are east, north and vertically upwards.
Forces \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) are given by \(\mathbf { p } = \left( \begin{array} { r } - 1 \\ - 1 \\ 5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1 \\ - 4 \\ 2 \end{array} \right) \mathrm { N }\) and \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 5 \\ 0 \end{array} \right) \mathrm { N }\).

1. Find which of \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) has the greatest magnitude.
2. A particle has mass 0.4 kg . The forces acting on it are \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and its weight. Find the magnitude of the particle's acceleration and describe the direction of this acceleration.

1 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are east, north and vertically upwards.
Forces \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) are given by \(\mathbf { p } = \left( \begin{array} { r } - 1 \\ - 1 \\ 5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1 \\ - 4 \\ 2 \end{array} \right) \mathrm { N }\) and \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 5 \\ 0 \end{array} \right) \mathrm { N }\).

1. Find which of \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) has the greatest magnitude.
2. A particle has mass 0.4 kg . The forces acting on it are \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and its weight. Find the magnitude of the particle's acceleration and describe the direction of this acceleration.