A particle is subjected to two simple harmonic motions along X-axis while other is along a line making angle 45° with the X-axis. The two motions are given by x = x₀ sin ωt and s =s₀ sin ωt. The amplitude of resultant motion is

x₀ + s₀ + 2x₀s₀

$\sqrt{x_{0}^{2} + s_{0}^{2}}$

$\sqrt{x_{0}^{2} - s_{0}^{2} + 2 x_{0} s_{0}}$

${[x_{0}^{2} + s_{0}^{2} + \sqrt{2} x_{0} s_{0}]}^{\frac{1}{2}}$

Solution

${[x_{0}^{2} + s_{0}^{2} + \sqrt{2} x_{0} s_{0}]}^{\frac{1}{2}}$

Two SHMs are separated at angle 45° So, resultant amplitude is

A_R = $\sqrt{{(s_{0} + x_{0}^{2} \cos^{2} 45^{o})}^{2} + {(x_{0} \sin 45^{o})}^{2}}$

= $\sqrt{(s_{0}^{2} + x_{0}^{2} \cos^{2} 45^{o} + 2 x_{0} s_{0} \cos 45^{o} + x_{0}^{2} \sin^{2} 45^{o})}$

Since sin 45^o = $\frac{1}{\sqrt{2}}$ ⇒ sin² 45^o = $\frac{1}{2}$

cos 45^o = $\frac{1}{\sqrt{2}}$ ⇒ cos² 45^o = $\frac{1}{2}$

A_R = $\sqrt{(s_{0}^{2} + x_{0}^{2} (\frac{1}{2}) + 2 x_{0} s_{0} (\frac{1}{\sqrt{2}}) + x_{0}^{2} (\frac{1}{2}))}$

⇒ A_R = $\sqrt{s_{0}^{2} + x_{0}^{2} + \sqrt{2} x_{0} s_{0}}$

Some More Questions From Oscillations Chapter

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