Explanation Given: The angular velocity of B , ω B = 4 rad / s . The position vector directed from P to A , r P / A = 4 in . . The angle formed by mechanism , θ = 30 ° . Show the free body diagram of the mechanism as in Figure (1). Conclusion: Express the radius of the circular path of motion P . r P = r P / A tan θ (I) Substitute 4 in . for r P / A and 30 ° for θ in Equation (I). r P = 4 tan 30 ° = 2.309 in . Express the linear velocity of P in the fixed system. v P = − ω B ( r P ) j (II) Substitute 4 rad / s for ω B and 2 .309 in . for r P in Equation (II). v P = − 4 ( 2.309 ) j = { − 9.238 j } in . / s Express the acceleration vector of P in the fixed system. a P = − ω B 2 ( r P ) i (III) Substitute 4 rad / s for ω B and 2 .309 in . for r P in Equation (III). a P = − ( 4 ) 2 ( 2.309 ) i = { − 36.95 i } in . / s 2 Write the motion of moving reference. v A = 0 a A = 0 Ω = ω A = ω A k Ω ˙ = α A = α A k Here, acceleration vector of A is a A , angular velocity is Ω , angular acceleration is Ω ˙ , angular velocity of A is ω A and angular acceleration of A is α A . Write the motion of collar P with respect to moving reference. r P / A = { 4 j } in . ( v P / A ) x y z = ( υ P / A ) x y z j ( a P / A ) x y z = ( a P / A ) j Apply the relative velocity equation. v P = v A + Ω × r P / A + ( v P / A ) x y z (IV) Here, linear velocity of A is v A . Substitute { − 9.238 j } in . / s for v P , 0 for v A , ω A k for Ω , { 4 j } in . for r P / A and ( υ P / A ) x y z j for ( v P / A ) x y z in Equation (IV). − 9.238 j = 0 + ( ω A k ) × ( 4 j ) + ( υ P / A ) x y z j = ( 4 ω A ) i − ( υ P / A ) x y z j (V) Equate j component from Equation (V)

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.8, Problem 152P

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The angular velocity ωA and angular acceleration αA of wheel A .

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Chapter 16.8, Problem 151P

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The angular velocity ω A and angular acceleration α A of wheel A .

Solution Summary: The author shows the free body diagram of the mechanism as in Figure (1).

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The angular velocity ωA and angular acceleration αA of wheel A .

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