1.7 Conservation of Mass Flow Rate in Fluids

The last major concept introduced in this unit is flow rate. While Bernoulli’s equation and Toricelli’s theorem from the last section remind us of the conservation of energy, the equations and concepts we learn in this section are based on the conservation of mass.

The flow rate (f), as defined above, is the flow speed of the liquid (v) times the cross sectional area (A) of the container. 🌊

If we know the area at 2 points in a pipe and the velocity at one point, we can predict the velocity at the other point using the continuity equation.

Because the same amount of mass has to flow through the pipe in a time interval, flow rates at any 2 points have to be the same.

It is important to notice that this equation works best when the fluid is incompressible. The calculations, therefore, will be most accurate when dealing with fluids but will only provide approximations when working with gases.

A. We can always assume that the cross section area of a pipe is circular. Since area is correlated to radius by a factor of a square. That means if the radius is doubled the area will be quadruples and if the radius is tripled, the area will increase 9x fold.

Increasing the radius by 1.5 means we are increasing the area by (1.5)^2 = 2.25

Since the flow rate is constant, we have to decrease the velocity by a factor of 2.25 since the area increased by a factor of 2.25.

Therefore velocity at point B is

20/2.25 = 8.89m/s

The AP test won’t often ask you to calculate the flow rate, but it will ask you to compare the velocities at 2 different places in a pipe 💯. The part of the pipe with a bigger area will have smaller speed, and the part with the smaller area will have greater speed. Since velocity and pressure are related, we can relate pressure and area.

This relationship is the essence of these later key concepts ❤️.

Fluids tend to make up about 10% of the AP exam. The most commonly tested topics are the buoyant force, pressure, the relationship between area, pressure and velocity, and Bernoulli's equation.

Typically this unit is tested in conjunction with Unit 2: Thermal Physics, which we will learn next.