Question

Bernolli equation solved question

Answer 1

Pressure inside a pipe

Step 1 - Make a prediction. In the pipe shown in

Figure 9.24, is the pressure higher at point 2,

where the fluid flows fastest, or at point 1? The

fluid in the pipe flows from left to right.

Many people predict that the pressure is higher at

point 2, where the fluid is moving faster.

Step 2 - Apply the continuity equation, and Bernoulli’s equation, to rank points 1, 2, and 3

according to pressure, from largest to smallest. Let’s see if the common prediction, that the

pressure is highest at point 2, is correct. First, apply the continuity equation: .

Looking at the tube, we know that , which tells us that .

Now, let’s apply Bernoulli’s Equation. Comparing points 1 and 2, we start with:

.

The vertical positions of these two points are equal so the terms cancel out:

.

Let’s re-write this as: .

The continuity equation told us that , so the right-hand side of the above equation

is positive. This means the left-hand side must also be positive, implying that Thus, the

pressure at point 2, where the fluid speed is highest, is less than the pressure at point 1. For points

at the same height, higher speed corresponds to lower pressure. We can make sense of this by

considering a parcel of fluid that moves from point 1 to point 2. Because this parcel of fluid

speeds up as it travels from point 1 to point 2, there must be a net force acting on it that is

directed right. This force must come from a difference in pressure between points 1 and 2. For the

force to be directed right, the pressure must be larger on the left, at point 1.

We can also use Bernoulli’s equation to show that the pressure at point 3 is equal to that

at point 1. Thus we can conclude that .