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Chapter 8: Problem 27

If an airplane propeller of radius \(5.0 \mathrm{ft}(=1.5 \mathrm{~m})\) rotates at 2000 rev/min and the airplane is propelled at a ground speed of \(300 \mathrm{mi} / \mathrm{h}(=480 \mathrm{~km} / \mathrm{h})\), what is the speed of a point on the tip of the propeller, as seen by \((a)\) the pilot and \((b)\) an observer on the ground? Assume that the plane's velocity is parallel to the propeller's axis of rotation.

Short Answer

Expert verified
The speed of a point on the tip of the propeller, as seen by (a) the pilot is 314.16 m/s and (b) an observer on the ground is 448.27 m/s.

Step by step solution

01

Convert the propeller rotations to velocity

First, we need to find out the speed of the tip of the propeller from the pilot's perspective. This is equal to the tangential speed, which is controlled by the rotation speed. The tangential speed \(v_t\) is calculated by the formula \(v_t = r * \omega\), where \(r\) is the radius and \(\omega\) is the angular speed. Let's convert 2000 revolutions per minute to radians per second: \( \omega = 2000 \, rev/min * (2\pi \, rad/rev) * (1 \, min/60 \, s) = 209.44 \, rad/s\). Therefore, the speed of the tip of the propeller from the pilot's perspective is \( v_t = 1.5 \, m * 209.44 \, rad/s = 314.16 \, m/s\).
02

Convert the plane's velocity

The plane's velocity needs to be converted from miles per hour to meters per second, since we're asked to provide the solution in the same units as the speed of the propeller tip. Therefore, \(v = 300 \, mi/h * (1609.34 \, m/mi) * (1 \, h/3600 \, s) = 134.11 \, m/s\).
03

Calculate the speed of the propeller tip as seen by an observer on the ground

Finally, to find the speed of the propeller tip from the perspective of an observer on the ground, we need to realize that the observer on the ground also sees the airplane's velocity. So, the total speed of the tip of the propeller, as seen by an observer on the ground, is the sum of the plane's speed and the tip's speed relative to the pilot: \(v_t' = v + v_t = 134.11 \, m/s + 314.16 \, m/s = 448.27 \, m/s\).

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