A frigate bird is soaring in a horizontal circular path. Its...

A frigate bird is soaring in a horizontal circular path. Its bank angle is estimated to be $25^{\circ}$ and it takes $13 \mathrm{~s}$ for the bird to complete one circle. ( $a$ ) How fast is the bird flying? ( $b$ ) What is the radius of the circle? (See "The Amateur Scientist" by Jearl Walker, Scientific American, March 1985, p. $122 .$.)

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Key Concepts

Period and Angular Velocity

The period is the time taken to complete one full revolution along a circular path. It relates to the angular velocity, defined as ? = 2?/T, where T is the period. The linear speed of the object can be determined using the relation v = ?R, linking the time it takes to make a complete turn, the angular velocity, and the radius of the circular motion.

Banked Flight Dynamics

Banked flight dynamics deals with how an object, such as an aircraft or bird, can turn by tilting its wings at an angle relative to the horizontal. In a banked turn, the lift force is divided into vertical and horizontal components; the vertical component counteracts gravity while the horizontal component provides the necessary centripetal force for the turn. This decomposition is fundamental in analyzing turns in flight.

Circular Motion

Circular motion involves an object moving along the circumference of a circle. A key aspect of circular motion is the centripetal acceleration, which is directed towards the center of the circle and is given by the formula a = v²/R, where 'v' is the speed of the object and 'R' is the radius of the circle. This concept is crucial for understanding how an object sustains its circular path.

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