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

11th Edition

ISBN: 9781259633133

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

الثانية Babakt Momentum equation for Boundary Layer S SS -Txfriction dray Momentum equation for Boundary Layer What laws are important for resolving issues 2 How to draw. 3 What's Point about this.

R αι g The system given on the left, consists of three pulleys and the depicted vertical ropes. Given: ri J₁, m1 R = 2r; απ r2, J2, m₂ m1; m2; M3 J1 J2 J3 J3, m3 a) Determine the radii 2 and 3.

B: Solid rotating shaft used in the boat with high speed shown in Figure. The amount of power transmitted at the greatest torque is 224 kW with 130 r.p.m. Used DE-Goodman theory to determine the shaft diameter. Take the shaft material is annealed AISI 1030, the endurance limit of 18.86 kpsi and a factor of safety 1. Which criterion is more conservative? Note: all dimensions in mm. 1 AA Motor 300 Thrust Bearing Sprocket 100 9750 เอ

Chapter 19 Solutions

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

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 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