Flowing Fluids for AP Physics B
Practice problems for these concepts can be found at:
Let's consider fluid is moving through a pipe. There are two questions that then arise: first, how do we find the velocity of the flow, and second, how do we find the fluid's pressure? To answer the first question, we'll turn to the continuity principle, and to answer the second, we'll use Bernoulli's equation.
Consider water flowing through a pipe. Imagine that the water always flows such that the entire pipe is full. Now imagine that you're standing in the pipe1—how much water flows past you in a given time?
We call the volume of fluid that flows past a point every second the volume flow rate, measured in units of m3/s. This rate depends on two characteristics: the velocity of the fluid's flow, v, and the cross-sectional area of the pipe, A. Clearly, the faster the flow, and the wider the pipe, the more fluid flows past a point every second. To be mathematically precise,
- Volume flow rate = Av.
If the pipe is full, then any volume of fluid that enters the pipe must eject an equal volume of fluid from the other end. (Think about it—if this weren't the case, then the fluid would have to compress, or burst the pipe. Once either of these things happens, the principle of continuity is void.) So, by definition, the volume flow rate is equal at all points within an isolated stream of fluid. This statement is known as the principle of continuity. For positions "1" and "2" in Figure 19.2, for example, we can write A1v1 = A2v2.
The most obvious physical consequence of continuity is that where a pipe is narrow, the flow is faster. So in Figure 19.2, the fluid must move faster in the narrower section labeled "1." Most people gain experience with the continuity principle when using a garden hose. What do you do to get the water to stream out faster? You use your thumb to cover part of the hose opening, of course. In other words, you are decreasing the cross-sectional area of the pipe's exit, and since the volume flow rate of the water can't change, the velocity of the flow must increase to balance the decrease in the area.
Bernoulli's equation is probably the longest equation you need to know for the AP exam, but fortunately, you don't need to know how to derive it. However, what you should know, conceptually, is that it's really just an application of conservation of energy.
Bernoulli's equation is useful whenever you have a fluid flowing from point "1" to point "2." The flow can be fast or slow; it can be vertical or horizontal; it can be open to the atmosphere or enclosed in a pipe. (What a versatile equation!) P is the pressure of the fluid at the specified point, y is the vertical height at the specified point, ρ is the fluid's density, and v is the speed of the flow at the specified point.
Too many terms to memorize? Absolutely not… but a mnemonic device might still be helpful. So here's how we remember it. First, we're dealing with pressures, so obviously there should be a pressure term on each side of the equation. Second, we said that Bernoulli's equation is an application of energy conservation, so each side of the equation will contain a term that looks kind of like potential energy and a term that looks kind of like kinetic energy. (Note that the units of every term are N/m2.)
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