Explanation Write the expression to obtain the relative velocity ( v r e l ) . v r e l = β ˙ × r A / O (I) Here, angular velocity of the motor is β ˙ . Write the expression to obtain the velocity of A ( v A ) . v A = v O + Ω × r A / O + v r e l (II) Write the expression to obtain the relative acceleration ( a r e l ) . a r e l = β ˙ × β ˙ × r A / O a r e l = β ˙ × v r e l (III) Write the expression to obtain the acceleration of A ( a A ) . a A = a O + Ω ˙ × r A + Ω × Ω × r A / O + 2 Ω × v r e l + a r e l (IV) Conclusion: Substitute b sin β j + b cos β k for r A / O in Equation (I). v r e l = − β ˙ i × ( b sin β j + b cos β k ) v r e l = − β ˙ b sin β k + b β ˙ cos β j v r e l = b β ˙ ( cos β j − sin β k ) Write the expression to obtain the velocity at O ( v O ) . v O = Ω × R j v O = Ω k × R j v O = − Ω R i Substitute − Ω R i for v O , b sin β j + b cos β k for r A / O and b β ˙ ( cos β j − sin β k ) for v r e l in Equation (II). v A = − Ω R i + Ω k ( b sin β j + b cos β k ) + b β ˙ ( cos β j − sin β k ) v A = − Ω R i − Ω b sin β i + b β ˙ ( cos β j − sin β k ) v A = − Ω ( R + b sin β ) i + b β ˙ cos β j − b β ˙ sin β k Thus, the velocity of center A of the ball is − Ω ( R + b sin β ) i + b β ˙ cos β j − b β ˙ sin β k

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ISBN: 9781119047315

Author: Bolton

Publisher: VST

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Chapter 7.6, Problem 45P

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The velocity and acceleration of center A of the ball.

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Chapter 7.6, Problem 44P

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