Three forces acting on a body are shown in the figure. To have the...

Three forces acting on a body are shown in the figure. To have the resultant force only along the

y

-direction, the magnitude of the minimum additional force needed is Three forces acting on a body are shown in the figure. To have the resultant force only along the y -direction, the

Three forces acting on a body are shown in the figure. To have the resultant force only along the y -direction, the

Options

A $0.5 \mathrm{~N}$
B $1.5 \mathrm{~N}$
C $\frac{\sqrt{3}}{4} \mathrm{~N}$
D $\sqrt{3} \mathrm{~N}$

Correct answer

A. $0.5 \mathrm{~N}$

Step-by-step solution

Minimum additional force needed

\begin{aligned} & F=-\left(F_{\text{resultant }}\right)_x \\ & F_{\text{resultant }}=[(4-2)(\cos 30 \hat{\mathbf{j}}-\sin 30 \hat{\mathbf{i}}) \\ = & {\left[2\left(\frac{\sqrt{3}}{2} \hat{\mathbf{j}}-\frac{1}{2} \hat{\mathbf{i}}\right)+\left(\frac{1}{2} \hat{\mathbf{i}}+\frac{\sqrt{3}}{2} \hat{\mathbf{j}}\right)\right] } \\ = & {\left[\left(\sqrt{3}+\frac{\sqrt{3}}{2}\right) \hat{\mathbf{j}}+\sin 60 \hat{\mathbf{j}}\right) } \\ = & {\left.\left[-\frac{1}{2} \hat{\mathbf{i}}+\frac{3 \sqrt{3}}{2} \hat{\mathbf{i}}+\frac{\hat{\mathbf{i}}}{2}\right)\right] } \\ = & -\frac{\hat{\mathbf{i}}}{2}+\frac{3 \sqrt{3}}{2} \hat{\mathbf{j}} \quad F=-\left(-\frac{\hat{\mathbf{i}}}{2}\right)=\frac{1}{2} \hat{\mathbf{i}}\end{aligned}

Hence,

|F|=0.5 \mathrm{ N}

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