Explanation Given: The radius of the propeller P , gears A and B is shown in Figure (1). The angular acceleration α is ( 300 t ) rad/s 2 . The radius of the gear A is 0.7 in . . The radius of the gear B is 1.4 in . . The radius of the propeller P is 2.20 in . . The time t is 0.75 s . Write the equation for the angular acceleration. α A = ( 300 t ) rad/s 2 (I) Write the expression for the angular acceleration. α = d ω d t d ω = α d t (II) Here, angular acceleration is α and rate of change of angular velocity with respect to time is ( d ω d t ) . Since, gear B is in mesh with gear A . Write the expression angular velocity ω B of gear B and ω A of gear A . ω A r A = ω B r B ω B = r A r B ω A (III) Here, angular velocity of gear A is ω A , angular velocity of gear B is ω B , radius of gear B is r B and radius of gear A is r A . Write the expression angular acceleration α B of gear B and α A of gear A . α A r A = α B r B α B = r A r B α A (IV) Here, angular acceleration of gear A is α A , angular acceleration of gear B is α B , radius of gear B is r B and radius of gear A is r A . Write the expression for the magnitude of the velocity. υ P = ω B r P (V) Here, velocity is υ A , angular velocity ω and radius is r A . Write the expression of tangential acceleration. a t = α B r P (VI) Here, tangential acceleration is ( a A ) t , angular acceleration is α , and radius is r A . Write the expression of normal acceleration. a n = ω B 2 r P (VII) Here, normal acceleration is ( a A ) n , angular velocity is ω and radius is r A . Write the expression for the magnitude of acceleration. a P = ( a t 2 ) + ( a n 2 ) (VIII). Here, acceleration is a B , tangential acceleration is a t and normal acceleration is a n . Conclusion: Substitute ( 300 t ) rad/s 2 for α in Equation (II). Integrate the Equation (II) at the limits 0 to ω A for ω and 0 to 0.75 s for t . ∫ 0 ω A d ω = ∫ 0 0.75 ( 300 t ) rad/s 2 d t [ ω ] 0 ω A = [ 200 t 3 / 2 ] 0 0

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.3, Problem 28P

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The magnitude of the velocity and acceleration of point P at tip of the propeller when time t is 0.75 s .

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Chapter 16.3, Problem 27P

Chapter 16.3, Problem 29P

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The magnitude of the velocity and acceleration of point P at tip of the propeller when time t is 0.75 s .

Solution Summary: The magnitude of velocity and acceleration of point P at the tip of the propeller when time t is 0.75 is shown in Figure (1).

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The magnitude of the velocity and acceleration of point P at tip of the propeller when time t is 0.75 s .

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