Explanation Given information: The weight ( W ) of the gyroscope rotor is 4 oz. The period ( τ 1 ) is 6 sec when the gyroscope rotor is 4 oz. The period ( τ 2 ) is 3.80 sec when the diameter ( d ) of the steel sphere is 1.25 in . . Assume the acceleration due to gravity ( g ) is 32.2 ft/s 2 . The specific gravity ( γ ) of the steel is 490 lb/ft 3 . Calculation: Show the free body diagram as in Figure (1). Refer Figure (1), Take moment about center of the system for external forces. ( ∑ M ) e x t e r n a l = − K θ Take moment about center of the system for effective forces. ( ∑ M ) e f f e c t i v e = I ¯ θ ¨ Equate the moment about O in the system for external and effective forces. ( ∑ M ) e x t e r n a l = ( ∑ M ) e f f e c t i v e − K θ = I ¯ θ ¨ − I ¯ θ ¨ − K θ = 0 I ¯ ( θ ¨ + K θ I ¯ ) = 0 θ ¨ + K θ I ¯ = 0 (1) Compare the differential Equation (1) with the general differential equation of motion ( x ¨ + ω n 2 x = 0 ) and express the natural circular frequency of vibration: ω n 2 = K I ¯ = K I ¯ Write the expression for period ( τ ) of oscillation: τ n = 2 π ω n Substitute K I ¯ for ω n . τ = 2 π K I ¯ = 2 π I ¯ K (2) Rewrite the above expression for the moment of inertia ( I ¯ ) . I ¯ = K τ 2 4 π 2 (3) Rewrite the above expression for the torsional spring constant ( K ) . K = 4 π 2 I ¯ τ 2 (4) Calculate the radius ( r ) using the relation: r = d 2 Substitute 1.25 in . for d . r = 1.25 2 = 0.625 in . × 1 ft 12 in . = 52.0833 × 10 − 3 ft Calculate the volume ( V s ) of the sphere: V s = 4 3 π r 3 Substitute 52.0833 × 10 − 3 ft for r . V s = 4 3 π ( 52.0833 × 10 − 3 ft ) 3 = 591.81 × 10 − 6 ft 3 Calculate the weight ( W s ) of the sphere using the relation: W s = γ V s Substitute 490 lb/ft 3 for γ and 591.81 × 10 − 6 ft 3 for V s

12th Edition

ISBN: 9781264095032

Author: BEER

Publisher: MCGRAW-HILL HIGHER EDUCATION

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 19.67P

To determine

The centroidal radius of gyration (k¯) of the rotor.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.2, Problem 19.66P

Chapter 19.2, Problem 19.68P

Students have asked these similar questions

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.

The inner rim of an 85-lb flywheel is placed on a knife edge, and the period of its small oscillations is found to be 1.26 s. Determine the centroidal moment of inertia of the flywheel.

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

VECTOR MECHANIC

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 centroidal radius of gyration ( k ¯ ) of the rotor.

Solution Summary: The author explains the centroidal radius of gyration of the rotor, the acceleration due to gravity, and the specific gravity.

Concept explainers

Videos

The centroidal radius of gyration (k¯) of the rotor.

Want to see the full answer?

Chapter 19 Solutions