Angular and linear velocity propeller

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SUMMARY

The discussion centers on calculating the maximum possible radius of an airplane propeller designed to operate at 2400 rpm while maintaining a tip speed below 270 m/s. The forward airspeed of the plane is 75 m/s, and the relationship between the tip velocity, plane velocity, and tangential velocity is defined by the equation vtip² = vplane² + vtan². Understanding the derivation of this equation is crucial for solving the problem, particularly in recognizing how the vector sum of the plane's forward motion and the propeller's tangential motion contributes to the overall tip velocity.

PREREQUISITES
  • Understanding of angular velocity and its relation to tangential velocity
  • Familiarity with vector addition in physics
  • Knowledge of basic kinematics and motion equations
  • Ability to manipulate and solve quadratic equations
NEXT STEPS
  • Study the derivation of the equation vtip² = vplane² + vtan² in detail
  • Learn about the effects of tip speed on propeller design and noise production
  • Explore the relationship between angular velocity and linear velocity in rotating systems
  • Investigate the principles of aerodynamics related to propeller performance
USEFUL FOR

Aerospace engineers, physics students, and anyone involved in the design and analysis of propeller systems will benefit from this discussion.

henry3369
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Homework Statement


You are designing an airplane propeller that is to turn at 2400 rpm (Fig. 9.13a). The forward airspeed of the plane is to be 75m/s, and the speed of the tips of the propeller blades through the air must not exceed 270m/s. (This is about 80% of the speed of sound in air. If the speed of the propeller tips were greater than this, they would produce a lot of noise.) What is the maximum possible propeller radius?

Homework Equations


vtip2 = vplane2 + vtan2

The Attempt at a Solution


I know how to solve this equation to find the maximum radius, but I'm having trouble understanding the derivation of the equation. I know that vplane points in the direction of the plane's motion; thus, the propeller also has this same velocity in the same direction. Then angular velocity can be related to tangential velocity by v = rw. How does the vector sum of these two result in the velocity of the tip of the propeller and isn't the direction of vtip also be tangential to the direction of motion of the propeller so vtip = vtan?
 
Physics news on Phys.org
Tip trajectory is a spiral in the air. The 75 m/s does contribute (a little) in the square root of the squared sum.
 

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