Explanation Given information: The radius of the cylinder is r . Calculation: Show the position of the particle inside the cylindrical surface as in Figure (1). Take the datum at position 2. For position 1: Write the expression for the displacement ( x ): x = R ( 1 − cos θ m ) Write the expression for kinetic energy ( T 1 ) : T 1 = 0 Write the expression for potential energy ( V 1 ) : V 1 = W x Substitute R ( 1 − cos θ m ) for x . V 1 = W R ( 1 − cos θ m ) ( 1 − cos θ m = 2 sin 2 θ m 2 ) = W R sin 2 θ m 2 For small oscillation sin θ ≈ θ . V 1 = W R θ m 2 2 For position 2. Write the expression maximum velocity ( v m ) : v m = R θ ˙ m Here, θ ˙ m is maximum angular velocity. Write the expression for kinetic energy ( T 2 ) : T 2 = 1 2 m v m 2 Substitute R θ ˙ m for v m . T 2 = 1 2 m ( R θ ˙ m ) 2 Write the expression for potential energy ( V 2 ) : V 2 = 0 Write the expression for weight ( W ): W = m g Write the expression for maximum angular velocity ( θ ˙ m ) :

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 19.3, Problem 19.72P

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The period of small (τn) oscillation of a small particle which moves without friction inside a cylindrical surface of radius R.

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Chapter 19.3, Problem 19.71P

Chapter 19.3, Problem 19.73P

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Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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The period of small ( τ n ) oscillation of a small particle which moves without friction inside a cylindrical surface of radius R .

Solution Summary: The author calculates the period of small oscillation of a small particle which moves without friction inside the cylindrical surface of radius R.

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The period of small (τn) oscillation of a small particle which moves without friction inside a cylindrical surface of radius R.

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Chapter 19 Solutions