$$(R - h)^2 + r^2 = R^2$$
$$R^2 + h^2 - 2Rh + r^2 = R^2$$
$$h^2 + r^2 = 2Rh$$
$$\dfrac{h^2 + r^2}{2h} = R$$ ___(i)
$$r >>> h$$
$$R = \dfrac{r^2}{2h}$$
$$\because h = \dfrac{w^2 r^2}{2g}$$
$$R = \dfrac{r^2 2g}{2w^2 r^2}$$
$$R = \dfrac{g}{w^2}$$ ___(ii)
$$\dfrac{\mu_1}{V} - \dfrac{\mu_2}{u} = \dfrac{\mu_1 - \mu_2}{R}$$
$$\dfrac{1}{V} + \dfrac{4}{3u} = \dfrac{1 - \dfrac{4}{3}}{R}$$
$$\dfrac{1}{V} = -\left(\dfrac{1}{3R} + \dfrac{4}{3(H - h)}\right)$$
$$\dfrac{1}{V} =-\left[\dfrac{1}{3R} + \dfrac{4}{3H}\right]$$
Put the value of equation (ii) $$h <<H$$
$$\dfrac{1}{V} = -\left[\dfrac{4}{3H} \left[1 + \dfrac{3H}{4} \dfrac{w^2}{39} \right]\right]$$
$$V = -\left[\dfrac{3H}{4} \left(1 + \dfrac{w^2 H}{4}\right)^{-1}\right]$$