Two persons A and B perform same amount of work in moving a body through a certain distance d with application of forces acting at angle 45º and 60º with the direction of displacement respectively. The ratio of force applied by person A to the force applied by person B is $\frac{1}{\sqrt{\mathrm{x}}}$. The value of x is ....... .

Given, work done by both person is same,

           ${\mathrm{W}}_{\mathrm{A}}={\mathrm{W}}_{\mathrm{B}}$

Direction of first force with displacement, θ₁ = 45°

Direction of second force with displacement, θ₂ = 60°

Ratio of force applied by person A to force applied by person B is ${\mathrm{F}}_{\mathrm{A}} : {\mathrm{F}}_{\mathrm{B}}=1 : \sqrt{\mathrm{x}}.$

Work done by both person is same,

${\mathrm{W}}_{\mathrm{A}}={\mathrm{W}}_{\mathrm{B}}\Rightarrow {\mathrm{F}}_{\mathrm{A}}\mathrm{dcos}{\mathrm{\theta }}_1={\mathrm{F}}_{\mathrm{B}}\mathrm{dcos}{\mathrm{\theta }}_2$

($\because$ distance is same for both = d)

$\begin{array}{l} \frac{{\mathrm{F}}_{\mathrm{A}}}{{\mathrm{F}}_{\mathrm{B}}}=\frac{\cos {\mathrm{\theta }}_1}{\cos {\mathrm{\theta }}_2}=\frac{\cos {60}^{\circ }}{\cos {45}^{\circ }}\\ \Rightarrow \frac{{\mathrm{F}}_{\mathrm{A}}}{{\mathrm{F}}_{\mathrm{B}}}=\frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}}=\frac{1}{\sqrt{2}}\Rightarrow \frac{{\mathrm{F}}_{\mathrm{A}}}{{\mathrm{F}}_{\mathrm{B}}}=\frac{1}{\sqrt{\mathrm{x}}}=\frac{1}{\sqrt{2}}\Rightarrow \mathrm{x}=2\end{array}$

Thus, the value of x is 2.