Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

Problem 34

Chapter 6, Problem 34

A medium-sized jet has a 3.8-m-diameter fuselage and a loaded mass of 85,000 kg. The drag on an airplane is primarily due to the cylindrical fuselage, and aerodynamic shaping gives it a drag coefficient of 0.37. How much thrust must the jet's engines provide to cruise at 230 m/s at an altitude where the air density is 1.0 kg/m³

Verified step by step guidance

Step 1: Identify the formula for drag force. The drag force is given by the equation:

F

_drag=1/2*C_d*ρ*A*v2, where

C

_d is the drag coefficient,

ρ

is the air density,

A

is the cross-sectional area, and

v

is the velocity.

Step 2: Calculate the cross-sectional area of the fuselage. The fuselage is cylindrical, so its cross-sectional area is the area of a circle:

A = π * {\frac{d}{2}}^{2}

, where

d

is the diameter of the fuselage. Substitute

d = 3.8

m into the formula.

Step 3: Substitute the given values into the drag force formula. Use

C

_d=0.37,

ρ = 1.0

kg/m³,

A

(calculated in Step 2), and

v = 230

m/s.

Step 4: Solve for the drag force

F

_drag. This represents the force opposing the motion of the jet due to air resistance. The engines must provide a thrust equal to this drag force to maintain cruising speed.

Step 5: Interpret the result. The thrust required to cruise at 230 m/s is equal to the drag force calculated in Step 4. This ensures the jet maintains constant velocity by balancing the drag force with the engine thrust.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Drag Force

Drag force is the resistance experienced by an object moving through a fluid, such as air. It is influenced by the object's shape, size, and speed, as well as the fluid's density. The drag force can be calculated using the formula: F_d = 0.5 * C_d * A * ρ * v^2, where F_d is the drag force, C_d is the drag coefficient, A is the frontal area, ρ is the fluid density, and v is the velocity.

Thrust

Thrust is the force that propels an aircraft forward, generated by its engines. To maintain a constant cruising speed, the thrust must equal the drag force acting on the aircraft. In this scenario, calculating the required thrust involves determining the drag force at the given speed and altitude, ensuring that the engines provide sufficient power to overcome this resistance.

Aerodynamic Coefficient

The aerodynamic coefficient, specifically the drag coefficient (C_d), quantifies the drag per unit area of an object in a fluid flow. It is a dimensionless number that reflects the object's shape and flow characteristics. A lower C_d indicates a more aerodynamically efficient shape, which is crucial for reducing drag and improving fuel efficiency in aircraft design.

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