Problem 62 - Fundamentals of Physics

Question

A pitot tube (Fig. \(14-48\) ) is used to determine the airspeed of an airplane. It consists of an outer tube with a number of small holes \(B\) (four are shown) that allow air into the tube; that tube is connected to one arm of a U-tube. The other arm of the U-tube is connected to hole \(A\) at the front end of the device, which points in the direction the plane is headed. At \(A\) the air becomes stagnant so that \(v_{A}=0 .\) At \(B,\) however, the speed of the air presumably equals the airspecd \(v\) of the plane. (a) Use Bernoulli's equation to show that $$ v=\sqrt{\frac{2 \rho g h}{\rho_{\text {ar }}}} $$ where \(\rho\) is the density of the liquid in the \(U\) -tube and \(h\) is the difference in the liquid levels in that tube. (b) Suppose that the tube contains alcohol and the level difference \(h\) is \(26.0 \mathrm{~cm}\). What is the plane's speed relative to the air? The density of the air is \(1.03 \mathrm{~kg} / \mathrm{m}^{3}\) and that of alcohol is \(810 \mathrm{~kg} / \mathrm{m}^{3}\)

Step-by-Step Solution

Answer

The plane's speed is approximately 197.6 m/s.

Step 1: Understanding the Problem

A Pitot tube is used to measure the airspeed of an airplane by comparing the stagnation pressure at point A (where speed is zero) and the dynamic pressure at point B (where speed is equivalent to the speed of the airplane). We will use Bernoulli's equation to derive the formula for the airspeed of the plane.

Step 2: Applying Bernoulli's Equation

Bernoulli's equation is given by: \( P + \frac{1}{2} \rho_{\text{air}} v^2 + \rho_{\text{air}} gh = \text{constant} \). Considering points A and B: \( P_A + 0 = P_B + \frac{1}{2} \rho_{\text{air}} v^2 \), since there is no height difference in the air part of the system.

Step 3: Relating Pressure Differences

The difference in pressure \( P_B - P_A \) causes a liquid level difference \( h \) in the U-tube. This pressure difference is also \( \rho g h \) where \( \rho \) is the liquid density. Equating it with the pressure difference in the air, we get: \( \frac{1}{2} \rho_{\text{air}} v^2 = \rho g h \).

Step 4: Solving for Airspeed

From the equation \( \frac{1}{2} \rho_{\text{air}} v^2 = \rho g h \), solve for \( v \): \( v = \sqrt{\frac{2 \rho g h}{\rho_{\text{air}}}} \). This gives us the expression for the airspeed of the airplane in terms of measurable quantities.

Step 5: Substituting Given Values

Substitute the given values into the formula: \( \rho_{\text{liquid}} = 810 \ \text{kg/m}^3 \), \( g = 9.81 \ \text{m/s}^2 \), \( h = 0.26 \ \text{m} \), and \( \rho_{\text{air}} = 1.03 \ \text{kg/m}^3 \).

Step 6: Calculating the Plane's Speed

Using the formula, calculate \( v \): \[ v = \sqrt{\frac{2 \times 810 \times 9.81 \times 0.26}{1.03}} \approx 197.6 \ \text{m/s} \].

Key Concepts

Pitot Tube

The Pitot tube is a fundamental device widely used in aviation and fluid dynamics to measure the velocity of a fluid flowing past the tube. It measures the airspeed of aircraft by tapping into both the static and dynamic pressure of the air. Typically, a Pitot tube consists of an outer tube with multiple small holes that connect to a U-tube manometer. The U-tube's fluid level difference provides a means to calculate the airspeed.

In operation, air enters the Pitot tube and stagnates at the front end (point A), where the velocity is zero. This stagnation leads to an increase in pressure, called stagnation pressure. The pressure at the other point, where the air moves freely past the tube's holes (point B), is the dynamic pressure and reflects the airplane's airspeed. By reading the pressure difference, one can determine the speed of air over the aircraft.

Airspeed Measurement

Measuring airspeed accurately is crucial for aircraft safety and efficiency. The Pitot tube uses the differential pressure to measure airspeed. This is derived from the principle that the change in pressure can be related to the change in velocity of the fluid.

In practical terms, when the airplane moves, the speed of the air relative to the aircraft increases. This increased speed results in an increase in pressure at the Pitot tube, observable by the rise in the U-tube's liquid level. The formula derived from Bernoulli's equation provides a clear relationship between the height difference in the fluid and airspeed. The algebraic manipulation leads us to use this pressure difference to calculate airspeed using the relationship:

• \[ v = \sqrt{\frac{2 \rho g h}{\rho_{\text{air}}}} \]

where \( \rho \) is the density of the fluid in the U-tube, \( g \) is the acceleration due to gravity, \( h \) is the fluid height difference, and \( \rho_{\text{air}} \) is the air density.

Fluid Dynamics

Fluid dynamics plays a pivotal role in understanding how the Pitot tube functions. It examines how fluids, including gases and liquids, move and how the forces within the fluid relate to its motion. Bernoulli's principle is a critical element in fluid dynamics, explaining the conservation of mechanical energy in a streamlining fluid.

In the context of the Pitot tube, Bernoulli's equation states that the sum of the pressure energy, kinetic energy, and potential energy per unit volume remains constant. This means that as the fluid's speed increases (from point A to B in the tube), its pressure decreases. The relationship can be simplified as:

• \[ P_A + \frac{1}{2} \rho_{\text{air}} v^2 = P_B \]

Realizing these principles aids in understanding how changes in speed and pressure interact within the Pitot tube system.

Pressure Difference

Pressure difference, the driving force behind the Pitot tube's function, is critical in the context of airspeed measurement. When air flows over the Pitot tube, the pressure at point A is greater due to stagnation compared to the dynamic pressure at point B.

It's this difference that is crucial for determining airspeed. The U-tube manometer connected to the Pitot tube exploits this pressure variation to calculate the elevation difference of the liquid inside. This height difference, correlated with the density of the fluid and the force of gravity, provides tangible data to calculate the airspeed via the equation:

• \[ h = \frac{(P_B - P_A)}{\rho g} \]

Mastering the concept of pressure difference is essential for anyone looking to understand how Pitot tubes provide vital data for pilots and engineers alike.