Two simple harmonic motion, are represented by the equations \({y}_{1}=10 \sin \left(3 \pi {t}+\frac{\pi}{3}\right)\) \(y_{2}=5(\sin 3 \pi t+\sqrt{3} \cos 3 \pi t)\) Ratio of amplitude of \({y}_{1}\) to \({y}_{2}={x}: 1\). The value of \({x}\) is ...... .

\({y}_{2}=5(\sin 3 \pi {t}+\sqrt{3} \cos 3 \pi {t})\)

\({y}_{2}=10\left(\frac{1}{2} \sin 3 \pi {t}+\frac{\sqrt{3}}{2} \cos 3 \pi {t}\right)\)

\({y}_{2}=10\left(\cos \frac{\pi}{3} \sin 3 \pi {t}+\sin \frac{\pi}{3} \cos 3 \pi {t}\right)\)

\({y}_{2}=10 \sin \left(3 \pi {t}+\frac{\pi}{3}\right) \Rightarrow \text { Amplitude }=10\)

So ratio of amplitudes \(=\frac{10}{10}=1\)