Two forces of magnitudes 2F and √2F act such that the resultant force is √10 F. Then find the angle between the two forces.

The correct answer is option 1) i.e. 45∘ 

CONCEPT:

• Triangle law of vector addition: It states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.

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The magnitude and direction of the resultant vector are calculated as follows:

\(\vec{R} =\vec{P} + \vec{Q}\)

The magnitude of R = \(\sqrt{P^2 + Q^2 +2PQcosθ}\)

The direction of the resultant, \(\phi = tan^{-1}(\frac{Qsinθ}{P + Qcosθ})\)

CALCULATION:

Given that:

The magnitude of the resultant R = √10F

Let the two forces be P = 2F and Q = √2F

The angle between the two forces is θ.

\(\sqrt{P^2 + Q^2 +2PQcosθ} = R\)

\(\sqrt{(2F)^2 + (√2 F)^2 +2(2F)(√2 F)cosθ} = √10 F\)

\(⇒ 4√2F^2cosθ = 4F^2\)

\(⇒ cosθ = \frac{1}{\sqrt2}\)

⇒ θ = 45∘