Question

Forces $X,Y,$ and $Z$ have magnitudes $10N,5(\sqrt{3}-1)N$ and $5(\sqrt{3}+1)N$, respectively. The forces $Y$ and $Z$ act in the same direction as shown in the figure. The resultant of $X$ and $Y$ and the resultant of $X$ and $Z$ have the same magnitude. Find $\theta$, the angle between $X$ and $Y$.

• A. 500
• B. 1000
• C. 150
• D. 0

Answer 1

The correct option is C 150

$\left|X\right|=10\left|Y\right|=5(\sqrt{3}-1)N\left|Z\right|=5(\sqrt{3}+1)N$

${\left|R_{XY}\right|}^2=X^2+Y^2+2XY\cos \theta$

${\left|R_{XZ}\right|}^2=X^2+Z^2+2XZ\cos \theta$

$\Rightarrow Y^2|+|H{Y}^2-Z^2=2X\cos \theta (Z-Y)$

$(Y-Z)(Y+Z)=2X\cos \theta (Z-Y)$

$5\sqrt{3}-5+5\sqrt{3}+5=2\times 10\cos \theta$

$-10\sqrt{3}=2\times 10\cos \theta$

$\Rightarrow \frac{-\sqrt{3}}{2}=\cos \theta$

$\therefore$ $\theta =180-{\cos }^{-1}\frac{\sqrt{3}}{2}=180-30={150}^0$