A mass is acted upon by three forces (as shown in the diagram below). Express the resultant force using the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the horizontal and vertical directions, and hence calculate the size of the resultant force acting on the mass. Describe the direction of this resultant force. (4 marks)
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\(\text{Resolving forces into \(\mathbf{i}\) and \(\mathbf{j}\) components}\)
\(\text{Resultant}\)
\(\text{Force:}\)
| \(F\) | \(=(-20+15 \cos 45+10 \cos 60) \mathbf{i}+(15 \sin 45-10 \sin 60)\mathbf{j}\) |
| \(=\left(\dfrac{15 \sqrt{2}}{2}-15\right)\mathbf{i}+\left(\dfrac{15 \sqrt{2}}{2}-5 \sqrt{3}\right)\mathbf{j}\) | |
| \(\abs{F}\) | \(=\sqrt{\left(\dfrac{3 \sqrt{2}}{2}-15\right)^2+\left(\dfrac{5 \sqrt{2}}{2}-5 \sqrt{3}\right)^2} =4.805 \cdots=4.81 \ N\) |
\(\text{Direction}\):
| \(\tan \theta\) | \(=\dfrac{\frac{15 \sqrt{2}}{2}-5 \sqrt{3}}{\frac{15 \sqrt{2}}{2}-15}=-0.443 \ldots\) |
| \(\theta\) | \(=180-23.89=156^{\circ}\) |
\(\text{i.e.} \ 156^{\circ} \ \text{direction measured from the \(x\)-axis}\)