Explanation Given information: The radius of the disk is r . The weight of the disk is W . When β is zero the period of small oscillation is given as τ 0 . Calculation: Show the free body diagram equation as in Figure (1). The angle α is θ ¨ . Write the expression for tangential acceleration ( a t ) : a t = r α Substitute θ ¨ is α . a t = r θ ¨ Write the expression for moment of inertia ( I ¯ D ) using the relation: I ¯ D = 1 2 W g r 2 Here, g is the acceleration due to gravity. The external forces in the system are tension in the belt and force due to the mass of the disk. The effective couple in the system is I ¯ θ ¨ . Take moment about C for external force, ∑ M C = w r sin ( β − θ ) − w r sin ( β + θ ) Take moment about C for effective force, ∑ ( M C ) e f f = 2 w g r a t + I ¯ α Equate the moment about C in the system for external and effective forces. ∑ M C = ∑ ( M C ) e f f w r sin ( β − θ ) − w r sin ( β + θ ) = 2 w g r a t + I ¯ α w r [ sin ( β − θ ) − sin ( β + θ ) ] = 2 w g r a t + I ¯ α ( ∵ cos A sin B = 1 2 sin ( A + B ) + sin ( A − B ) ) − ( 2 w r sin θ cos β ) = 2 w g r a t + I ¯ α ( ∵ sin θ ≈ 0 ) − ( 2 w r cos β ) θ = 2 w g r a t + I ¯ α Substitute θ ¨ for α , 1 2 W g r 2 for I ¯ D and r θ ¨ for a t

12th Edition

ISBN: 9781265566296

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 19.2, Problem 19.44P

To determine

The angle β for which the period of small oscillation is 2τ0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.2, Problem 19.43P

Chapter 19.2, Problem 19.45P

Students have asked these similar questions

Two 40-g weights are attached at A and B to the rim of a 1.5-kg uniform disk of radius r = 100 mm. Determine the frequency of small oscillations when β = 60°.

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 1300-kg sports car has a center of gravity G located a distance h above a line connecting the front and rear axles. The car is suspended from cables that are attached to the front and rear axles as shown. Knowing that the periods of oscillation are 4.04 s when L = 4 m and 3.54 s when L = 3 m, determine h and the centroidal radius of gyration.

Chapter 19 Solutions

<LCPO> VECTOR MECH,STAT+DYNAMICS

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 angle β for which the period of small oscillation is 2 τ 0 .

Solution Summary: The author shows the free body diagram equation as in Figure (1). The angle is beta and the weight of the disk is W.

Concept explainers

Videos

The angle β for which the period of small oscillation is 2τ0.

Want to see the full answer?

Chapter 19 Solutions