Explanation Given information: The radius of the half section cylinder is r . The mass of the cylinder is m . The radius of the cylinder A and B is r 4 . The mass of the cylinder is m 4 . Calculation: Show the position 1 and position 2 of the system with velocity as in Figure (1). For position 1: Write the expression for the displacement ( h ): h = 4 r 3 π ( 1 − cos θ m ) = 4 r 3 π ( 1 − cos θ m ) ( ∵ ( 1 − cos θ m = 2 sin 2 θ m 2 ) ) = 4 r 3 π 2 sin 2 θ m 2 For small oscillation 2 sin 2 θ m 2 ≈ θ m 2 2 . h = 4 r 3 π θ m 2 2 = 2 r θ m 2 3 π Write the expression for the potential energy ( V 1 ) : V 1 = m g h Substitute 2 r θ m 2 3 π for h . V 1 = 2 m g r θ m 2 3 π Write the expression for the kinetic energy ( T 1 ) : T 1 = 0 For position 2: Calculate the mass moment of inertia of the cylinder A ( I A ) and B ( I B ) using the relation: I A = I B = 1 2 ( m 8 ) ( r 4 ) 2 = m r 2 256 Calculate the mass moment of inertia of the half section cylinder ( I h ) using the relation: I h = 1 2 m r 2 The natural frequency of the cylinder A ( ω A ) and B ( ω B ) : ω A = ω B = 4 ω Write the expression for the kinetic energy ( T 2 ) : T 2 = 1 2 I A ω A 2 + 1 2 I B ω B 2 + 1 2 I h Substitute m r 2 256 for I A , m r 2 256 for I B , 4 ω for ω A , 4 ω for ω B , and 1 2 m r 2 for I h

12th Edition

ISBN: 9781260265453

Author: BEER

Publisher: MCG

See similar textbooks

Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

Videos

Question

Chapter 19.3, Problem 19.92P

To determine

The frequency (fn) of small oscillations.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.3, Problem 19.91P

Chapter 19.3, Problem 19.93P

Students have asked these similar questions

A 14-oz sphere A and a 10-oz sphere C are attached to the ends of a rod AC of negligible weight that can rotate in a vertical plane about an axis at B. Determine the period of small oscillations of the rod.

A connecting rod is supported by a knife-edge at point A; the period of its small oscillations is observed to be 0.87 s. The rod is then inverted and supported by a knife edge at point B and the period of its small oscillations is observed to be 0.78 s. Knowing that ra + rb=10 in. determine(a) the location of the mass center G. (b) the centroidal radius of gyration k.

A period of 6.00 s is observed for the angular oscillations of a 4-oz gyroscope rotor suspended from a wire as shown. Knowing that a period of 3.80 s is obtained when a 1.25-in.-diameter steel sphere is suspended in the same fashion, determine the centroidal radius of gyration of the rotor. (Specific weight of steel = 490 lb/ft3.)

Chapter 19 Solutions

VECTOR MECH...,STAT.+DYN.(LL)-W/ACCESS

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - A particle moves in simple harmonic motion....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 - A simple pendulum consisting of a bob attached to...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 frequency ( f n ) of small oscillations.

Solution Summary: The author shows the position 1 and position 2 of the system with velocity as in Figure (1).

Concept explainers

Videos

The frequency (fn) of small oscillations.

Want to see the full answer?

Chapter 19 Solutions