Two forces P and Q have magnitudes 2F and 3F respectively, acting at an angle theta to each other. If the magnitude of force Q is doubled, the resulting force also doubles. What is the value of the angle theta?

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EEJ MAIN Physics QUESTION #7018

Question 1

Two forces $P$ and $Q$ have magnitudes $2F$ and $3F$ respectively, acting at an angle $\theta$ to each other. If the magnitude of force $Q$ is doubled, the resulting force also doubles. What is the value of the angle $\theta$?

$120^\circ$✔️
$60^\circ$
$90^\circ$
$30^\circ$

Correct Answer Explanation

Using the resultant formula $R^2 = A^2 + B^2 + 2AB \cos \theta$. Initially, $R^2 = (2F)^2 + (3F)^2 + 2(2F)(3F)\cos \theta = 13F^2 + 12F^2 \cos \theta$. When $Q$ is doubled ($6F$), the new resultant is $2R$, so $(2R)^2 = (2F)^2 + (6F)^2 + 2(2F)(6F)\cos \theta = 40F^2 + 24F^2 \cos \theta$. Substituting $4R^2$ and solving the equations yields $\cos \theta = -0.5$, which corresponds to $120^\circ$.

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