Explanation Given: The expression for the radial coordinate is shown in Figure (1). The angular velocity ( θ ˙ ) is 2 rad/s . The transverse coordinate ( θ ) is 30 ° . The angular acceleration ( θ ¨ ) is 3 rad/s 2 . From Figure (1) write the expression for the radial co-ordinate ( r ) . r = 4 sec θ ft (I). Write the expression for the radial acceleration. a r = r ¨ − r θ ˙ 2 (II). 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 ˙ (III). Here, transverse acceleration is a θ and second time derivative of θ is θ ¨ and first time derivative of radius ( r ) is r ˙ . Write the expression for the magnitude of the acceleration. a = a r 2 + a θ 2 (IV). The magnitude of the acceleration is a , radial acceleration is a r and transverse acceleration a θ . Write the expression radial velocity. υ r = r ˙ (V). Here, radial velocity is υ r and first time derivative of radius ( r ) is r ˙ . Write the expression transverse velocity. υ θ = r θ ˙ (VI). Here, transverse velocity is υ θ , radius of the particle is r and angular velocity is θ ˙ . Write the expression for the magnitude of the velocity. υ = υ r 2 + υ θ 2 (VII). Here, the magnitude of the acceleration is υ , radial velocity is υ r and the transverse velocity is υ θ . Conclusion: Differentiate Equation (I). r = 4 sec θ ft r ˙ = ( 4 sec θ ( tan θ ) ) θ ˙ ft/s (VIII). Substitute 30 ° for θ in Equation (I). r = 4 sec θ ft = 4 sec ( 30 ) = 4.619 ft Differentiate Equation (VIII). r ˙ = ( 4 sec θ ( tan θ ) ) θ ˙ ft/s r ¨ = 4 [ sec θ ( tan θ ) θ ¨ + θ ˙ ( sec θ ( sec 2 θ ) θ ˙ + tan θ sec θ ( tan θ ) θ ˙ ) ] r ¨ = 4 [ sec θ ( tan θ ) θ ¨ + θ ˙ 2 ( sec 3 θ + tan 2 θ sec θ ) ] ft/s 2 (IX). Substitute 30 ° for θ and 2 rad/s for θ ˙ in Equation (VIII). r ˙ = ( 4 sec θ ( tan θ ) ) θ ˙ ft/s r ˙ = ( 4 sec ( 30 ) ( tan ( 30 ) ) ) ( 2 ) ft/s r ˙ = 5.333 ft/s Substitute 30 ° for θ , 3 rad/s for θ ¨ and 2 rad/s for θ ˙ and in Equation (IX)

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 12.8, Problem 182P

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The magnitude of velocity and acceleration of the peg when (θ) is 30° .

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Chapter 12.8, Problem 181P

Chapter 12.8, Problem 183P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The magnitude of velocity and acceleration of the peg when ( θ ) is 30 ° .

Solution Summary: The author explains the expression for the radial co-ordinate (stackreldottheta).

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The magnitude of velocity and acceleration of the peg when (θ) is 30° .

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Chapter 12 Solutions