Explanation Given: The velocity of B , υ B = 7 m / s . The velocity of A , υ A = 8 m / s . The distance between A and B , r B / A = 2 m . The radius of circular path, r = 1 m . The angular acceleration or speed of B , α B = 4 m / s 2 . The angular acceleration or speed of A , α A = 6 m / s 2 . Conclusion: Express the angular velocity of particle A . Ω = υ A r (I) Substitute 8 m / s for υ A and 1 m for r in Equation (I). Ω = 8 1 = 8 rad / s = { − 8 k } rad / s Express the relative velocity of particle B with respect to A by applying relative velocity equation. v B = v A + Ω × r B / A + ( v B / A ) x y z (II) Substitute { 7 i } m / s for v B , { − 8 i } m / s for v A , { − 8 k } rad / s for Ω and { 2 j } m for r B / A in Equation (II). 7 i = − 8 i + ( − 8 k ) × ( 2 j ) + ( v B / A ) x y z 7 i = − 8 i − 16 i + ( v B / A ) x y z ( v B / A ) x y z = { 31 } i m / s Hence, the relative velocity of B with respect to A is { 31 i } m / s . Express the angular acceleration due to the rotation of x , y , z reference. Ω ˙ = α A r (III) Substitute 6 m / s 2 for α A and 1 m for r in Equation (III). Ω ˙ = 6 1 = 6 rad / s 2 = { − 6 k } rad / s 2 Express the normal component acceleration of A . ( a A ) n = ( υ A ) 2 r (IV) Substitute 8 m / s for υ A and 1 m for r in Equation (IV). ( a A ) n = ( 8 ) 2 1 = 64 m / s 2 Write the given differential equation for parabolic path. y = x 2 (V) Differentiate Equation (V) with respect to x to get, y = x 2 d y d x = 2 x = 0 (VI) Perform double differentiation in Equation (VI) to get, d 2 y d x 2 = 2 Express the radius of the parabolic path. ρ = [ 1 + ( d y / d x ) 2 ] 3 / 2 | d 2 y d x 2 | (VII) Substitute 0 for d y d x and 2 for d 2 y d x 2 in Equation (VII). ρ = [ 1 + ( 0 ) 2 ] 3 / 2 | 2 | = 1 2 m = 0

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Author: HIBBELER

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Chapter 16, Problem 137P

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Chapter 16, Problem 136P

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