Explanation Draw the schematic diagram of the rod OA and the ball as shown in figure (1). Write the expression for the sliding distance of the ball. R = 200 + 50 sin 2 π n t mm (I). Substitute 2 for n in equation (I). R = 200 + 50 sin 2 π ( 2 ) t mm R = 200 + 50 sin 4 π t mm (II). Differentiate the ϕ values to get the velocity and angular acceleration of the ϕ component. ϕ ˙ = 0 ϕ ¨ = 0 Differentiate the equation (II) with respect to time. R ˙ = 50 × 4 π cos 4 π t = 200 π cos 4 π t (III). Again differentiate the equation (III). R ¨ = − 200 π × 4 π sin 4 π t R ¨ = − 800 π 2 sin 4 π t (IV). Write the expression for the radial acceleration. a R = R ¨ − R ϕ ˙ 2 − R θ ˙ 2 cos 2 ϕ (V). Write the expression for the tangential acceleration. a θ = cos ϕ R ⋅ d d t ( R 2 θ ˙ ) − 2 R θ ˙ ϕ sin ϕ a θ = cos ϕ R ⋅ 2 R ⋅ R ˙ θ ˙ + cos ϕ R ⋅ R 2 ⋅ θ ¨ − 2 R θ ˙ ϕ ˙ sin ϕ a θ = 2 R ˙ θ ˙ cos ϕ − 2 R θ ˙ ϕ ˙ sin ϕ (VI). Write the expression for the transverse acceleration. a ϕ = 1 R ⋅ d d t ( R 2 ϕ ˙ ) + R θ ˙ 2 sin ϕ cos ϕ (VII). Conclusion: Convert the angular rate rev/min to rad/s. θ ˙ = 120 rev min × 2 π 60 = 4 π rad/s Differentiate the angular rate gives angular acceleration. d d t θ ˙ = 4 π rad/s θ ¨ = 0 Calculate the angle ϕ . ϕ = 90 ° − 30 ° = 60 ° Refer the equation (IV), The velocity is maximum when cos 4 π t = 1 and R is constant. For this condition, sin 4 π t = 0 Substitute 0 for sin 4 π t in equation (II). R = 200 + 50 ( 0 ) mm R = 200 mm Substitute 0 for sin 4 π t in equation (IV). R ¨ = − 800 π 2 ( 0 ) = 0 Substitute 0 for R ¨ , 200 mm for R , 0 for ϕ ˙ , 4 π rad/s for θ ˙ , and 60 ° for ϕ in equation (V)

8th Edition

ISBN: 9781118885840

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

Publisher: WILEY

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Chapter 2.7, Problem 178P

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