(a) Explanation Given information: The angle is β = 10 ° . The sphere S , a uniform cylinder C , and a thin pipe P . The time as t = 4 s . Calculation: The acceleration due to gravity as g = 9.81 m / s 2 or 32.2 ft / s 2 . Calculate the acceleration ( a ¯ ) as shown below. a ¯ = r α (1) Here, α is the angular acceleration. Calculate the mass moment of inertia ( I ¯ ) as shown below. I ¯ = m k ¯ 2 Here, m is the mass and k ¯ is the radius of gyration. The radius of gyration of pipe is k ¯ = r . The radius of gyration of cylinder is k ¯ = r 2 . The radius of gyration of sphere is k ¯ = 2 5 r 2 . Sketch the Free Body Diagram as shown in Figure 1. Refer to Figure 1. Apply the Equilibrium of moment about C as shown below. ∑ M C = I G α + m a d ( W sin β ) r = ( m a ¯ ) r + I ¯ α Substitute r α for a ¯ and m k ¯ 2 for I ¯ . ( m g sin β ) r = ( m r α ) × r + m k ¯ 2 α r g sin β = ( r 2 + k ¯ 2 ) α α = r g sin β r 2 + k ¯ 2 Calculate the acceleration ( a ¯ ) as shown below. Substitute r g sin β r 2 + k ¯ 2 for α in Equation (1). a ¯ = r r g sin β r 2 + k ¯ 2 = r 2 r 2 + k ¯ 2 g sin β (2) For pipe. Substitute r for k ¯ in Equation (2). a P = r 2 r 2 + r 2 g sin β = r 2 2 r 2 g sin β = 1 2 g sin β For cylinder. Substitute r 2 for k ¯ in Equation (2). a C = r 2 r 2 + ( r 2 ) 2 g sin β = r 2 r 2 + r 2 2 g sin β = r 2 3 2 r 2 g sin β = 2 3 g sin β For sphere (b) To determine The distance between the sphere and the cylinder.

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 16.2, Problem 16.95P

(a)

To determine

Find the distance between the pipe and the cylinder.

(b)

To determine

The distance between the sphere and the cylinder.

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Chapter 16.2, Problem 16.94P

Chapter 16.2, Problem 16.96P

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Chapter 16 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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Find the distance between the pipe and the cylinder.

Solution Summary: The author calculates the acceleration due to gravity as g=9.81 and the mass moment of inertia.

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Find the distance between the pipe and the cylinder.

The distance between the sphere and the cylinder.

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Chapter 16 Solutions