4 A small box has weight \(\mathbf { W } \mathrm { N }\) and is held in equilibrium by two strings with tensions \(\mathbf { T } _ { 1 } \mathrm {~N}\) and \(\mathbf { T } _ { 2 } \mathrm {~N}\). This situation is shown in Fig. 2 which also shows the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) that are horizontal and vertically upwards, respectively.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b80eced6-2fea-4b95-9104-d13339643df0-2_252_631_414_803}
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\caption{Fig. 2}
\end{figure}
The tension \(\mathbf { T } _ { 1 }\) is \(10 \mathbf { i } + 24 \mathbf { j }\).
- Calculate the magnitude of \(\mathbf { T } _ { 1 }\) and the angle between \(\mathbf { T } _ { 1 }\) and the vertical.
The magnitude of the weight is \(w \mathrm {~N}\).
- Write down the vector \(\mathbf { W }\) in terms of \(w\) and \(\mathbf { j }\).
The tension \(\mathbf { T } _ { 2 }\) is \(k \mathbf { i } + 10 \mathbf { j }\), where \(k\) is a scalar.
- Find the values of \(k\) and of \(w\).