3. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.]
Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\), are given by
$$\mathbf { F } _ { 1 } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 2 } = ( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { N } \quad \mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$$
where \(a\) and \(b\) are constants.
The forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a particle \(P\) of mass 4 kg .
Given that \(P\) rests in equilibrium on a smooth horizontal surface under the action of these three forces,
- find the size of the angle between the direction of \(\mathbf { F } _ { 3 }\) and the direction of \(- \mathbf { j }\).
The force \(\mathbf { F } _ { 3 }\) is now removed and replaced by the force \(\mathbf { F } _ { 4 }\) given by \(\mathbf { F } _ { 4 } = \lambda ( \mathbf { i } + 3 \mathbf { j } )\) N, where \(\lambda\) is a positive constant.
When the three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) act on \(P\), the acceleration of \(P\) has magnitude \(3.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
- Find the value of \(\lambda\).