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(C) For a simple harmonic motion, the displacement is given by $y = A \sin(\omega t)$ and velocity is $v = \frac{dy}{dt} = A \omega \cos(\omega t)$.Re

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$10 \ Hz$

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(C) For a simple harmonic motion, the displacement is given by $y = A \sin(\omega t)$ and velocity is $v = \frac{dy}{dt} = A \omega \cos(\omega t)$.Rearranging these, we get $\frac{y}{A} = \sin(\omega t)$ and $\frac{v}{A \omega} = \cos(\omega t)$.Squaring and adding, we get $\frac{v^2}{(A \omega)^2} + \frac{y^2}{A^2} = 1$, which represents an ellipse.The major axis along the $X$-axis (velocity) is $2A \omega$ and the minor axis along the $Y$-axis (displacement) is $2A$.The ratio of the major axis to the minor axis is $\frac{2A \omega}{2A} = \omega$.Given that the ratio is $20 \pi$, we have $\omega = 20 \pi$.Since $\omega = 2 \pi f$, we have $2 \pi f = 20 \pi$.Therefore, $f = 10 \ Hz$.

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