In the first how they got the RHS term .

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The RHS is the rate of change of flux $\phi=BA$ through the loop, which equals the induced emf (Faraday's Law) :

$V=-\frac{d\phi}{dt}=-B\dot A$

because $B$ is constant.

$\frac12 \pi a^2$ is the area of the semi-circular portion of the loop, which is the only part which changes. The projection of this semi-circle onto the plane of the loop is $A=\frac12 \pi a^2 \cos\theta$ where $\theta=\omega t$ is the angle between the semi-circle and the loop. Differentiating we get $\dot A=-\frac12\pi a^2 \omega \sin(\omega t)$.

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