Two equal forces to P and 2P act on a particle.If the first be doubled and the second be increased by 20N, the direction of the resultant is unaltered.Find the value of P.

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Solution

Let the resultant make angle $β$ with the force P.
$I n f i r s t c a s e, tan β = \frac{2 P sin θ}{P + 2 P cos θ} I n s e c o n d c a s e, tan β = \frac{(2 P + 20) sin θ}{2 P + (2 P + 20) cos θ} \frac{(2 P + 20) sin θ}{2 P + (2 P + 20) cos θ} = \frac{2 P sin θ}{P + 2 P cos θ} [∵ \frac{a}{b} = \frac{c}{d} = \frac{a - c}{b - d}] w e g e t 2 P = 20 P = 10 N$

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Two equal forces to P and 2P act on a particle.If the first be doubled and the second be increased by 20N, the direction of the resultant is unaltered.Find the value of P.

Let the resultant make angle β with the force P.Infirstcase,tanβ=2PsinθP+2PcosθInsecondcase,tanβ=(2P+20)sinθ2P+(2P+20)cosθ(2P+20)sinθ2P+(2P+20)cosθ=2PsinθP+2Pcosθ[∵ab=cd=a−cb−d]weget2P=20P=10N