Explanation Given: The radial distance and the angle is shown in Figure (1). The mass of collars m = 0.5 kg . The radial distance r = ( 1.6 cos θ ) m . The angular displacement is θ = 30 ° . The angular velocity θ ˙ = 4 rad/s . Write the expression for the radial distance ( r ) . r = ( 1.6 cos θ ) m (I). Write the expression for the angular velocity ( θ ˙ ) . θ ˙ = 4 rad/s (II). Write the expression for the radial acceleration. a r = r ¨ − r θ ˙ 2 (III). Here, radial acceleration is a r and second time derivative of radius ( r ) is r ¨ . Write the expression for the transverse acceleration. a θ = r θ ¨ + 2 r ˙ θ ˙ (IV). Here, transverse acceleration is a θ and second time derivative of θ is θ ¨ and first time derivative of radius ( r ) is r ˙ . Differentiate Equation (I). r = ( 1.6 cos θ ) m r ˙ = ( − 1.6 sin θ θ ˙ ) m / s 2 (V). Differentiate Equation (V). r ˙ = ( − 1.6 sin θ θ ˙ ) m / s r ¨ = − 1.6 cos θ θ ˙ 2 − 1.6 sin θ θ ¨ m / s 2 (VI). Differentiate Equation (V). θ ˙ = 4 rad/s θ ¨ = 0 Draw the free body diagram as shown in Figure (2). Refer Figure (2) write the formula for equation of motion. ∑ F r = m a r (VII). Here, summation of total force acting on radial direction is ∑ F r mass is m and acceleration radial direction is a r . Refer Figure (2) calculate the components of force. ∑ F r = − N c cos 45 ° − m ( 9.81 m/s 2 ) cos 45 ° (VIII). Refer Figure (2) write the formula for equation of motion. ∑ F θ = m a θ (IX). Here, summation of total force acting on transverse direction is ∑ F θ mass is m and acceleration in the transverse direction is a θ . Refer Figure (2) calculate the components of force. ∑ F θ = F O A − N c sin 45 ° − m ( 9.81 m/s 2 ) sin 45 ° (X). Conclusion: Substitute 45 ° for θ in Equation (I). r = ( 1.6 cos θ ) m r = ( 1.6 cos ( 45 ° ) ) m = 1.1314 m Substitute 45 ° for θ and 4 rad/s for θ ˙ in Equation (V). r ˙ = ( − 1.6 sin θ θ ˙ ) m / s r ˙ = ( − 1.6 sin 45 ° ( 4 ) ) m / s = − 4.5255 m/s Substitute 45 ° for θ , 4 rad/s for θ ˙ and 0 for θ ¨ in Equation (VI)

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 13.6, Problem 110P

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The normal force which the circular rod exerts on one of the collars and the force that OA exerts on the other collar when θ=45°

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Chapter 13.6, Problem 109P

Chapter 13.7, Problem 112P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The normal force which the circular rod exerts on one of the collars and the force that O A exerts on the other collar when θ = 45 °

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The normal force which the circular rod exerts on one of the collars and the force that OA exerts on the other collar when θ=45°

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Chapter 13 Solutions