Conservation of Mass Flow Rate in Fluids

#Fluid Dynamics: Conservation of Mass Flow Rate 🌊

Hey there, future AP Physics 2 master! Let's dive into the world of fluids and explore how mass is conserved when things are flowing. This is a crucial concept, and we'll make sure it sticks! This section is all about understanding how the speed of a fluid changes as it moves through different areas, and it's based on the principle of mass conservation. Let's get started!

# Flow Rate Basics

Flow rate (f) is simply the speed of the fluid (v) multiplied by the cross-sectional area (A) of the container. Think of it like this: how much water is passing a certain point in a pipe per second? The formula is:

$$f = vA$$

# The Continuity Equation

The continuity equation is a direct result of the conservation of mass. It states that the flow rate must be the same at any two points in a pipe. This is because the same amount of mass has to flow through the pipe in a given time interval. Here's the equation:

$$A_1v_1 = A_2v_2$$

#Example Problem: Changing Pipe Radius

Let's tackle a typical problem:

Q: A point in a pipe, A, has a radius of X meters and a liquid velocity of 20 m/s. Another point, B, in the same pipe has a radius of 1.5X. Find the velocity of the liquid at point B.

A:

1. Area and Radius: Remember that the cross-sectional area of a pipe is circular, so area (A) is related to radius (r) by $A = \pi r^2$. If the radius is doubled, the area quadruples, and if the radius is tripled, the area increases by a factor of 9. 2. Area Change: Increasing the radius by 1.5 means the area increases by $(1.5)^2 = 2.25$.
2. Constant Flow Rate: Since the flo...