Question

If. two equal forces of magnitude p act at an angle
2 θ, then their resultant will be __​

Answer 1

Given : two equal forces of magnitude p act at an angle 2θ,

To Find :  resultant

Solution:

Resultant = √A² + B² + 2AB Cosα

α is the angle between vectors A & B

A = p

B = p

α = 2θ

Resultant force =  √p² + p² + 2pp Cos2θ

= √2p² + 2p²Cos2θ

= √2p(√(1 + Cos2θ)

= √2p(√2Cos²θ)

= √2p √2  .Cos θ

= 2pCos θ

Resultant force  = 2pCos θ

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Answer 2

Given:

Two equal forces of magnitude p act at an angle 2θ.

To find:

Net resultant ?

Calculation:

Applying Equation of Vector Addition:

Let resultant be R :

$\therefore \: R = \sqrt{ {p}^{2} + {p}^{2} + 2(p)(p) \cos(2 \theta) }$

$\implies \: R = \sqrt{ {p}^{2} + {p}^{2} + 2 {p}^{2} \cos(2 \theta) }$

$\implies \: R = \sqrt{ 2{p}^{2}+ 2 {p}^{2} \cos(2 \theta) }$

$\implies \: R = \sqrt{ 2{p}^{2} \{1+ \cos(2 \theta) \}}$

$\implies \: R = \sqrt{ 2{p}^{2} \{1+ 2{\cos}^{2} (\theta) - 1\}}$

$\implies \: R = \sqrt{ 4{p}^{2} {\cos}^{2} (\theta) }$

$\implies \: R = 2p \cos( \theta)$

So, final answer is:

$\boxed{ \bf \: R = 2p \cos( \theta) }$