First slide

Vectors

Question

Five forces 2 N, $\sqrt{3} N, 5 N, \sqrt{3} and 2 N respectively act at a particle P as shown in the figure .$
The resultant force on the particle P is

Moderate

A

10 N making angle $60^{0} with X - axis$
B

$10 N making angle 60^{0} with Y - axis$
C

20 N along Y-axis
D

None of these

Solution

$\sum_{} F_{x} = sum of x - component of forces$
= $2 + \sqrt{3} \cos 30^{0} + 5 \cos 60^{0} - 2 \sin 30^{0}$
= $2 + \frac{3}{2} + \frac{5}{2} - 1 = 5 N$
$\sum_{} F_{y} = sum of y - component of forces$
= $\sqrt{3} \sin 30^{0} + 5 \sin 60^{0} + \sqrt{3} + 2 \cos 30^{0}$
= $\frac{\sqrt{3}}{2} + \frac{5 \sqrt{3}}{2} + \sqrt{3} + \sqrt{3} = 5 \sqrt{3} N$
$∴ F = \sqrt{{(\sum_{} F_{x})}^{2} + {(\sum_{} F_{y})}^{2}} = \sqrt{5^{2} + {(5 \sqrt{3})}^{2}}$
= $\sqrt{100} = 10 N$
$∴ tanθ = \frac{\sum_{} F_{y}}{\underset{}{\sum F_{x}}} = \frac{5 \sqrt{3}}{5} = \sqrt{3} \Rightarrow θ = 60^{0}$

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