Explanation Given information: The mass ( m A B ) of the slender rod AB is 3 kg. The mass ( m 0 ) of the disk is 5 kg. The spring constant ( k ) is 280 N/m . The length ( l ) of the rod is 300 mm. The radius ( r ) of the disk is 80 mm. Assuming the acceleration due to gravity ( g ) as 9.81 m/s 2 . Calculation: Show the position 1 and position 2 of the slender rod with disk as in the Figure 1. For position 1: Write the expression for mass moment of inertia of disk ( I d i s k ) : I d i s k = 1 2 m 0 r 2 Write the expression for mass moment of inertia of rod ( I A ) r o d : ( I A ) r o d = 1 3 m A B l 2 Express the term ( θ ˙ m ) : θ ˙ m = ω n θ m Write the expression for the kinetic energy ( T 1 ) : T 1 = 1 2 I d i s k θ ˙ m 2 + 1 2 ( I A ) r o d θ ˙ m 2 Substitute 1 2 m 0 r 2 for I d i s k and 1 3 m A B l 2 for ( I A ) r o d . T 1 = 1 2 ( 1 2 m 0 r 2 ) θ ˙ m 2 + 1 2 ( 1 3 m A B l 2 ) θ ˙ m 2 Substitute ω n θ m for θ ˙ m . T 1 = 1 2 ( 1 2 m 0 r 2 ) ω n 2 θ m 2 + 1 2 ( 1 3 m A B l 2 ) ω n 2 θ m 2 Write the expression for the potential energy ( V 1 ) : V 1 = 0 Position 2: Write the expression for the kinetic energy ( T 2 ) : T 1 = 0 Write the expression for the displacement ( y m ) for position 2: h = l 2 ( 1 − cos θ m ) ( ∵ ( 1 − cos θ m = 2 sin 2 θ m 2 ) ) = l 2 2 sin 2 θ m 2 For small oscillation 2 sin 2 θ m 2 ≈ θ m 2 2 . h = l 2 θ m 2 2 = l 4 θ m 2 Write the expression for the displacement ( h ) for position 2: x m = r θ m Write the expression for the potential energy ( V 2 ) : V 2 = 1 2 k ( x m ) 2 + m A B g h Substitute l 4 θ m 2 for h and r θ m for x 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

See similar textbooks

Concept explainers

Forced undamped vibrations

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

Videos

Question

Chapter 19.3, Problem 19.80P

To determine

The period of vibration (τn) of the system.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.3, Problem 19.79P

Chapter 19.3, Problem 19.81P

Students have asked these similar questions

An 8-kg uniform disk of radius 200 mm is welded to a vertical shaft with a fixed end at B. The disk rotates through an angle of 3° when a static couple of magnitude 50N.m is applied to it. If the disk is acted upon by a periodic torsional couple of magnitude Tm=60N.m. determine the range of values of vf for which the amplitude of the vibration is less than the angle of rotation caused by a static couple of magnitude Tm.

A 15-lb uniform cylinder can roll without sliding on an incline and is attached to a spring AB as shown. If the center of the cylinder is moved 0.4 in. down the incline and released, determine (a) the period of vibration, (b) the maximum velocity of the center of the cylinder.

A 3-kg slender rod AB is bolted to a 5-kg uniform disk. A spring of constant 280 N/m is attached to the disk and is unstretched in the position shown. If end B of the rod is given a small displacement and released, determine the period of vibration of the system.

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

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The period of vibration ( τ n ) of the system.

Solution Summary: The author shows the position 1 and position 2 of the slender rod with disk as in the Figure 1.

Concept explainers

Videos

The period of vibration (τn) of the system.

Want to see the full answer?

Chapter 19 Solutions