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Chapter 10: Problem 43

An eagle with 2.1 -m wingspan flaps its wings 20 times per minute, each stroke extending from \(45^{\circ}\) above the horizontal to \(45^{\circ}\) below. Downward and upward strokes take the same time. On a given down stroke, what's (a) the average angular velocity of the wing and (b) the average tangential velocity of the wingtip?

Short Answer

Expert verified
The average angular velocity of the wing during a downstroke is calculated from the total change in angle and the time taken for one stroke. This gives the angular speed in radians per hour. The radius, half the wingspan of the eagle, is used along with the angular speed to calculate the average tangential velocity of the wingtip in m/h.

Step by step solution

01

Calculate Change in Angle and Time

First, determine the total change in angle which will be from \(45^{\circ}\) above horizontal to \(45^{\circ}\) below. This makes a total of \(45^{\circ} +45^{\circ} = 90^{\circ}\). Convert this to radians as \( rad = \frac{90 * \pi}{180} \). For time, since the eagle flaps 20 times a minute, and a complete flap includes upward and downward strokes, which takes same time, the time taken for one down stroke will be \( \frac{1}{2} \times \frac{1}{20*60} \) hours.
02

Calculate Average Angular Velocity

Next, use the formula for angular velocity \(\omega = \frac{\Delta\theta}{\Delta t}\). Here, \(\Delta\theta = rad \) and \(\Delta t = \frac{1}{2} \times \frac{1}{20*60} \) hours. Plug in the values, you get the average angular velocity.
03

Calculate the Radius

The radius of the circle in which the wingtip moves is half of the wingspan, so \( r = \frac{2.1}{2} \) m.
04

Calculate Average Tangential Velocity

Now, use the relation \(v = r\omega\) where \(r = \frac{2.1}{2}\) m and \(\omega\) is the angular velocity calculated in step 2. This gives the average tangential velocity of the wingtip.

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