$y_{2}=5(\sin 3 \pi t+\sqrt{3} \cos 3 \pi t)$
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 $=\frac{10}{10}=1$
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| column $I$ | column $II$ |
| $(A)$ $U _1( x )=\frac{ U _0}{2}\left[1-\left(\frac{ x }{ a }\right)^2\right]^2$ | $(P)$ The force acting on the particle is zero at $x = a$. |
| $(B)$ $U _2( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2$ | $(Q)$ The force acting on the particle is zero at $x=0$. |
| $(C)$ $U _3( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2 \exp \left[-\left(\frac{ x }{ a }\right)^2\right]$ | $(R)$ The force acting on the particle is zero at $x =- a$. |
| $(D)$ $U _4( x )=\frac{ U _0}{2}\left[\frac{ x }{ a }-\frac{1}{3}\left(\frac{ x }{ a }\right)^3\right]$ | $(S)$ The particle experiences an attractive force towards $x =0$ in the region $| x |< a$. |
| $(T)$ The particle with total energy $\frac{ U _0}{4}$ can oscillate about the point $x=-a$. |
