To measure the airspeed of a plane, a device called a Pitot...

To measure the airspeed of a plane, a device called a Pitot tube is used. A simplified model of the Pitot tube is a manometer with one side connected to a tube facing directly into the "wind" (stopping the air that hits it head-on) and the other side connected to a tube so that the "wind" blows across its openings. If the manometer uses mercury and the levels differ by $25 \mathrm{cm},$ what is the plane's airspeed? The density of air at the plane's altitude is $0.90 \mathrm{kg} / \mathrm{m}^{3}$. (FIGURE CANT COPY)

Key Concepts

Dynamic Pressure

Dynamic pressure is the component of pressure that is associated with the motion of the fluid. It is mathematically expressed as one half the product of the fluid density and the square of its velocity. This concept is central to linking the measured pressure differential to the speed of the airflow.

Manometer

A manometer is an instrument that measures pressure differences by balancing the weight of a fluid column against the pressure exerted by a gas or liquid. In this context, the height difference of the mercury column is used to quantify the pressure difference, which can be directly related to the kinetic energy of the airflow.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at equilibrium due to the force of gravity. It is calculated by multiplying the fluid density, gravitational acceleration, and the height of the fluid column. In the context of a manometer, the difference in height of the fluid column provides a measure of the pressure difference between two points.

Pitot Tube

A Pitot tube is an instrument used to measure fluid velocity by converting the kinetic energy of the flow into potential energy. It works by capturing the stagnation pressure of the fluid—where the fluid is brought to rest—and comparing it with the static pressure of the fluid, allowing for the calculation of the flow speed.

Bernoulli's Principle

Bernoulli's Principle states that in a steady, incompressible flow, the sum of the static pressure, dynamic pressure, and gravitational potential energy per unit volume remains constant. This principle is key in determining the relationship between the pressure difference measured by the device and the velocity of the fluid.

Transcript

00:01 In this exercise, we have a mercury manometer that's used to measure the speed of an airplane.

00:05 The second column of the manometer is connected to a tube through which flows the wind created by the airplane's movement.

00:14 And the first column is just connected to still air.

00:18 And we want to calculate what is the speed of the airplane, given that the difference in height between the mercury in the two columns is 25 centimeters.

00:30 So what we're going to need is the density of the air that the exercise gave us.

00:36 That's 0 .9 kilograms per cubic meter.

00:39 And we're also going to need the density of mercury, which we can look up in a table, and that's 13 ,600 kilograms per cubic meter.

00:51 Okay.

00:53 And the first thing we're going to have to do, the first step, is to calculate the difference of pressure between the steel air and the wind.

01:02 In order to do that, we can just read in the manometer.

01:06 The difference of pressure is going to be row of the liquid inside the manometer, in this case is the density of the mercury, times g times the difference in height between the two columns, h.

01:26 And in the second step, we're going to use bernoulli's equation and the difference of pressure that we found in order to find the speed of the airplane.

01:38 So according to bernoulli's equation, the pressure p1 of the still air is going to be equal to the pressure of the wind, p2, plus row v2 squared over 2.

01:57 Now notice some things here...

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