Question

A child loves to watch as you fill a transparent bottle with shampoo. Every cross section is a circle, but the diameters of the circles have different values, so that the bottle is much wider in some place than others. You pour a bright green shampoo with constant volume flow rate 16.5 cubic meter/second. At what rate is its level in the bottle rising (a) at a point where the diameter of the bottle is 2.48 in and (b) at a point where the diameter is 13.5 mm?

Answer 1

$$Q=16.5\frac{cm^3}{s}=16.5\cdot10^{-6}\frac{m^3}{s}$$

a) We have:

$$Q=A_1v_1$$

$$v_1=\frac{Q}{A_1}=\frac{4Q}{\pi d_1^2}=\frac{4(16.5\cdot10^{-6})}{\pi (2.48\cdot 0.0254)^2}$$

$$v_1=0.00529\frac{m}{s}=5.29\frac{mm}{s}$$

b)

$$v_2=\frac{Q}{A_2}=\frac{4Q}{\pi d_2^2}=\frac{4(16.5\cdot10^{-6})}{\pi (13.5\cdot 10^{-3})^2}$$

$$v_2=0.115\frac{m}{s}=115\frac{mm}{s}$$