To determine Determine the values of v and v ˙ at the instant, the angle β and the radius of curvature of the path. Explanation Express the tangential velocity component of the airplane with respect to origin at O . v O θ = r θ ˙ (I) Here, the distance of the airplane from the origin is r and the rate of change of angular displacement of the radial vector is θ ˙ Express the radial velocity component of the airplane with respect to origin at O . v O r = r ˙ (II) The absolute velocity of airplane measured using the aixs at O must be equal to the tangential velocity of airplane about the point C . v a = v v = ( v O θ ) 2 + ( v O r ) 2 (III) Here, the absolute velocity of airplane measured using the aixs at O is v a and tangential velocity of airplane about the point C is v . Show the velocity polygon showing the radial and tangential velocity components and the absolute velocity of airplane as in Figure (1). Consider the velocity polygon. sin ( θ − β ) = v O r v (IV) Express the tangential acceleration component of the airplane with respect to axis at O . a O r = r ¨ − r θ ˙ 2 (V) Express the radial acceleration component of the airplane with respect to axis at O . a O θ = r θ ¨ + 2 r ˙ θ ˙ (VI) Show the acceleration vectors with respect to points O and C from the same point as in Figure (2). Express the absolute acceleration of airplane. a = ( a O r ) 2 + ( a O θ ) 2 (VII) Express the angle made by absolute acceleration with the horizontal. tan ( θ − α ) = a O θ a O r (VIII) Express the radial acceleration component with respect to C . a r = a cos ( α − β ) (IX) Express angular velocity. v ˙ = a sin ( α − β ) (X) Concluison: Substitute 90 m for r and 0 .53 rad / s for θ ˙ in Equation (I). v O θ = ( 90 m ) ( 0 .53 rad / s ) = 47.7 m / s Substitute 15 .5 m / s for r ˙ in Equation (II)

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 2.6, Problem 163P

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Determine the values of v and v˙ at the instant, the angle β and the radius of curvature of the path.

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Chapter 2.6, Problem 162P

Chapter 2.6, Problem 164P

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