Question 8

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $$d_1$$ and $$d_2$$ are filled in the tube. Each liquid subtends 90° angle at centre. Radius joining their interface makes an angle $$\alpha$$ with vertical. Ratio $$\frac{d_1}{d_2}$$ is:

$$h_1 = R[1 - \sin\alpha]; \quad h_2 = R[1 - \cos\alpha]$$

$$h_2^1 = R\sin\alpha + R\cos\alpha$$

$$h_1 d_1 g = h_2 d_1 g + h_2^1 d_2 g$$

$$d_1 R[1 - \sin\alpha] = d_1 R[1 - \cos\alpha] + d_2 R[\sin\alpha + \cos\alpha]$$

$$d_1 [\cos\alpha - \sin\alpha] = d_2 [\cos\alpha + \sin\alpha]$$

$$\frac{d_1}{d_2} = \frac{\cos\alpha + \sin\alpha}{\cos\alpha - \sin\alpha} = \frac{1 + \tan\alpha}{1 - \tan\alpha}$$

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