3/ Forces of magnitude 1, 2, 3, 4, 5 and 6N respectively act at the centre of a
regular hexagon towards its six angular points. Determine the magnitude and
direction of the resultant.
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the magnitude is 6N and direction of the resultant = 60° if forces of magnitude 1, 2, 3, 4, 5 and 6N respectively act at the centre of a
regular hexagon
Step-by-step explanation:
Let say a Hexagon ABCDEF with centre O.
Let us consider AB and AE as the axes.
Let say AB, BC,CD,DE,,EF and FA are of magnitudes 1,2,3,4, 5 and 6 respectively.
Let R be the resultant vector making an angle θ
Resolving the forces along Ax
R Cosθ=1+2 Cos 60° +3 cos120° -4-5 Cos 60° -6 cos 120°
=1+2.1/2 +3(-1/2)-4-5.1/2 -6(-1/2)
=1+1-3/2-4-5/2+3
=-3 N
Resolving along AY ,
we get
R sinθ=2 Sin60° +3 sin120°-5sin60° -6sin120°
=2sin60°+3 sin60° -5sin60°-6sin60°
=-6sin60°
=-6.√3/2
=-3√3 N
R sinθ=-3√3 N
R²Cos²θ + R²Sin²θ = (-3)² + (-3√3)²
=> R²(Cos²θ + Sin²θ) =9 +27
=> R²=36
=> R= 6N
Magnitude = 6N
R sinθ/RCosθ = -3√3 / -3
=> Tanθ = √3
=> θ = 60°
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