(a) Explanation Given information: The side ( a ) of the square is 300 mm. The constant angular velocity ( ω ) of the clevis is 12 rad/s. Calculation: Let point O be the midpoint of axle AB and consider principal axes x ′ , y ′ , and z ′ with origin at point O . Sketch the axes of the system as shown in Figure (1). Refer Figure (1), Write the equation of the mass moment of inertia about x ′ -axis. I x ′ = 1 12 m a 2 Write the equation of the mass moment of inertia about z ′ -axis. I z ′ = 1 3 m a 2 Find the principal moment of inertia about y ′ -axis using the equation: I y ′ = I x ′ + I z ′ Substitute 1 12 m a 2 for I x ′ and 1 3 m a 2 for I z ′ . I y ′ = 1 12 m a 2 + 1 3 m a 2 = 5 12 m a 2 Write the vector form of angular velocity ( ω ) of the system. ω = − ω sin β i ′ + ω cos β j ′ Write the equation of angular momentum ( H O ) about point O . H O = I x ′ ω x ′ i + I y ′ ω y ′ j + I z ω z k (1) Substitute − ω sin β i ′ for ω x ′ i , ω cos β j ′ for ω y ′ j , and 0 for ω z . H O = I x ′ ( − ω sin β i ′ ) + I y ′ ( ω cos β j ′ ) + I z ( 0 ) k = − I x ′ ω sin β i ′ + I y ′ ω cos β j ′ Write the angular velocity of reference frame ( Ω ) . Ω = ω Substitute − ω sin β i ′ + ω cos β j ′ for ω . Ω = − ω sin β i ′ + ω cos β j ′ The rate of change of angular momentum ( H ˙ O ) O x y z about the reference frame is 0. Because the reference frame rotating about fixed point O . Write the equation of rate of change of angular momentum ( H ˙ O ) of the system. H ˙ O = ( H ˙ O ) O x y z + Ω × H O Substitute 0 for ( H ˙ O ) O x y z , − ω sin β i ′ + ω cos β j ′ for Ω , and − I x ′ ω sin β i ′ + I y ′ ω cos β j ′ for H O (b) To determine The largest value of angular velocity ( ω ) for which the plate remains vertical.

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259977251

Author: BEER

Publisher: MCG

See similar textbooks

Concept explainers

Conservation of Energy

In physics, the rule of the conservation of energy states that energy doesn’t disappear from the atmosphere; instead, it changes its form from one type to another. Energy is one of the most important physical quantities in nature, and it cannot be create…

Videos

Question

Chapter 18.2, Problem 18.87P

(a)

To determine

The value of angular velocity (ω) for which the plate forms a constant angle β=60° with the horizontal x-axis.

(b)

To determine

The largest value of angular velocity (ω) for which the plate remains vertical.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 18.2, Problem 18.86P

Chapter 18.2, Problem 18.88P

Students have asked these similar questions

0.08 m,x 0.225 0.75 0.08 m. O: 1.08 b' 0.108 (c) Couple polygon. (d) Force polygon. Fig. 21.11 Page 18 of 19 Balarcing of Rotating Masses H.W. 2: A shaft carries five masses A, B, C, D and E which revolve at the same radius in planes which are equidistant from one another. The magnitude of the masses in planes A. Cand Dare 50 kg. 40 kg and 80 kg respectively. The angle between A and C is 90° and that between C and D is 135°, Determine the magnitude of the masses in planes B and E and their positions to put the shaft in complete rotating balance. 19 Page 19 uf 19 II

Two uniform cylinders, each of weight W = 14 lb and radius r = 5 in., are connected by a belt as shown. Knowing that at the instant shown the angular velocity of cylinder B is 30 rad/s clockwise, determine (a) the distance through which cylinder A will rise before the angular velocity of cylinder B is reduced to 5 rad/s, (b ) the tension in the portion of belt connecting the two cylinders.

Consider the mechanism shown. Members PQ and QR are joined by a hinge at Q. End P of member PQ is pin-supported and end R of member QR is constrained to move along a horizontal surface. Member PQ rotates clockwise at a constant rate of 12 rad/s. Member QR rotates counterclockwise at a rate of 3.84 rad/s. Which of the following gives the closest value to the magnitude of the angular acceleration of rod QR? 9.16, 6.18, 1.609, 35.2 rad/s^2?? Which of the following gives the closest value to the magnitude of the acceleration of point R? 3.13, 9.89, 10.28, 12.88 m/s^2??

Chapter 18 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 18.1 - A thin, homogeneous disk of mass m and radius r...Ch. 18.1 - Prob. 18.2P Ch. 18.1 - Prob. 18.3P Ch. 18.1 - A homogeneous disk of weight W = 6 lb rotates at...Ch. 18.1 - A homogeneous disk of mass m = 8 kg rotates at the...Ch. 18.1 - A solid rectangular parallelepiped of mass m has a...Ch. 18.1 - Prob. 18.8P Ch. 18.1 - Determine the angular momentum HD of the disk of...Ch. 18.1 - Prob. 18.10P Ch. 18.1 - Determine the angular momentum HO of the disk of...

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 value of angular velocity ( ω ) for which the plate forms a constant angle β = 60 ° with the horizontal x -axis.

Solution Summary: The author calculates the angular velocity and the mass moment of inertia using the following equations.