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Two persons A and B

Question:

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 $45^{\circ}$ and $60^{\circ}$ with the direction of displacement respectively. The ratio of force applied by person A to the force applied by person $B$ is $\frac{1}{\sqrt{x}}$. The value of $x$ is ........

Solution:

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

$\mathrm{F}_{\mathrm{A}} \mathrm{d} \cos 45^{\circ}=\mathrm{F}_{\mathrm{B}} \mathrm{d} \cos 60^{\circ}$

$\mathrm{F}_{\mathrm{A}} \times \frac{1}{\sqrt{2}}=\mathrm{F}_{\mathrm{B}} \times \frac{1}{2}$

$\frac{\mathrm{F}_{\mathrm{A}}}{\mathrm{F}_{\mathrm{B}}}=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$

$\mathrm{x}=2$