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Question

The resultant of two forces having magnitude 3p and 2p is R. If the force having magnitude 3p is doubled keeping the same direction, then the resultant is also doubled. The angle between two forces is

A
120o
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B
150o
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C
180o
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D
125o
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Solution

The correct option is A 120o
Let θ be the angle between two forces. We know resultant of two forces F1 and F2 inclined at an angle θ is given by
R=F21+F22+2F1F2cosθ
Or R2=F21+F22+2F1F2cosθ(1)
Here F1=3p and F2=2p
R2=(3p)2+(2p)2+2×3p×2p cosθ
or R2=9p2+4p2+12p2cosθ
R2=13p2+12p2cosθ(2)
Now F1=2×3p=6p, R=2R, F2=2p
From equation (1),
(2R)2=(6p)2+(2p)2+2×6p×2pcosθ
4R2=36p2+4p2+24p2cosθ
4R2=40p2+24p2cosθ
or R2=10p2+6p2cosθ(3)
equating (2) and (3), we get,
13p2+12p2cosθ=10p2+6p2cosθ
Or 3p2=6p2cosθcosθ=12
θ=cos1(12)=120
θ=120

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