Dependence of Liquid Spray Rate on Densities

To determine the dependence of the volumetric rate of liquid sprayed on the densities of air and liquid, we apply Bernoulli's principle.

1. Pressure Drop Created by the Air Flow: Let the speed of the piston be $v_p$, the cross-sectional area of the cylinder be $A_c$, and the cross-sectional area of the nozzle be $A_n$. By the equation of continuity for air (assuming it to be incompressible at low speeds): $A_c v_p = A_n v_a \implies v_a = \left(\frac{A_c}{A_n}\right) v_p$ For a given piston speed, the velocity of the air rushing through the nozzle, $v_a$, is constant.

Applying Bernoulli's principle to the airflow, the high-velocity air at the nozzle produces a localized reduction in pressure $P$ compared to the atmospheric pressure $P_{\text{atm}}$: $P_{\text{atm}} - P = \frac{1}{2} \rho_a v_a^2$

2. Flow of Liquid through the Tube: The surface of the liquid in the reservoir is exposed to the atmosphere at pressure $P_{\text{atm}}$, while the top end of the tube opening inside the nozzle experiences the reduced pressure $P$.

Applying Bernoulli's equation to the liquid drawn up the vertical tube of negligible or fixed height $h$: $P_{\text{atm}} = P + \rho_\ell g h + \frac{1}{2} \rho_\ell v_\ell^2$

Neglecting the hydrostatic term $\rho_\ell g h$ relative to the dynamic pressure, or considering the dynamic pressure difference driving the flow: $P_{\text{atm}} - P \approx \frac{1}{2} \rho_\ell v_\ell^2$

Equating the two expressions for the pressure difference $(P_{\text{atm}} - P)$: $\frac{1}{2} \rho_\ell v_\ell^2 = \frac{1}{2} \rho_a v_a^2$

Solving for the speed of the liquid $v_\ell$: $v_\ell = v_a \sqrt{\frac{\rho_a}{\rho_\ell}}$

3. Volumetric Flow Rate of the Liquid: The volumetric rate $\frac{dV}{dt}$ at which the liquid is sprayed out is given by: $\frac{dV}{dt} = A_{\text{tube}} v_\ell = A_{\text{tube}} v_a \sqrt{\frac{\rho_a}{\rho_\ell}}$

Since $A_{\text{tube}}$ and $v_a$ are constants for a predetermined piston speed, the rate is proportional to: $\frac{dV}{dt} \propto \sqrt{\frac{\rho_a}{\rho_\ell}}$

Therefore, the correct option is (A).