(a) Explanation Given information: The radius of a homogeneous disk ( a ) is 9 in.. The length of the rod AG ( l ) is 18 in.. The angle that the rod AG lies ( θ 0 ) is 90 ° . The angular velocity of the shaft AB ϕ ˙ m is twice its initial value ϕ ˙ 0 . Calculation: Consider Z axis be vertical. The principal moment of inertia for principal axes xyz with origin at A are I ′ = I x = I y = m [ 1 4 a 2 + ( 2 a ) 2 ] and I = I z = 1 2 m a 2 . Draw the Free body diagram of homogeneous disk and its vectors position as in Figure (1). Draw the FBD of dimensions of the disk as in Figure (2). Write the expression for the unit vector K. K = − i sin θ + k cos θ Write the expression for the angular momentum about point A . H A = I x ω x i + I y ω y j + I z ω z k Substitute I ′ for I x , ϕ ˙ sin θ for ω x , I ′ for I y , − θ ˙ for ω y , and I for I z . H A = I ′ ( ϕ ˙ sin θ ) i − I ′ θ ˙ j + I ω z k Determine the value of kinetic energy T . T = 1 2 I x ω x 2 + 1 2 I y ω y 2 + 1 2 I z ω z 2 Substitute I ′ for I x , ϕ ˙ sin θ for ω x , I ′ for I y , − θ ˙ for ω y , and I for I z . T = 1 2 I x ω x 2 + 1 2 I y ω y 2 + 1 2 I z ω z 2 = 1 2 I ′ ( ϕ ˙ sin θ ) 2 + 1 2 I ′ θ ˙ 2 + 1 2 I ω z 2 = 1 2 I ′ ( ϕ ˙ 2 sin 2 θ + θ ˙ 2 ) + 1 2 I ω z 2 Determine the angular velocity along z axis ω z using the relation. ω z = ψ ˙ + ϕ ˙ cos θ For constraint of clevis ψ ˙ = 0 . Substitute 0 for ψ ˙ . ω z = ϕ ˙ cos θ Determine the conservation of angular momentum about fixed Z axis H A ⋅ K . Substitute I ′ ( ϕ ˙ sin θ ) i − I ′ θ ˙ j + I ω z k for H A and ( i sin θ + k cos θ ) for K. H A ⋅ K = ( I ′ ( ϕ ˙ sin θ ) i − I ′ θ ˙ j + I ω z k ) . ( i sin θ + k cos θ ) = I ′ ϕ ˙ sin 2 θ + I ω z cos θ (1) Substitute α for H A ⋅ K , m [ 1 4 a 2 + ( 2 a ) 2 ] for I ′ , 1 2 m a 2 for I , and ϕ ˙ cos θ for ω z in Equation (1). α = m [ 1 4 a 2 + ( 2 a ) 2 ] ϕ ˙ sin 2 θ + 1 2 m a 2 ( ϕ ˙ cos θ ) cos θ = 17 4 m a 2 ϕ ˙ sin 2 θ + 1 2 m a 2 ϕ ˙ cos 2 θ Determine the value of conservation of energy using the relation (b) To determine The initial angular velocity ϕ ˙ 0 of shaft AB .

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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 18.3, Problem 18.135P

(a)

To determine

The minimum value of θm.

(b)

To determine

The initial angular velocity ϕ˙0 of shaft AB.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 18.3, Problem 18.134P

Chapter 18.3, Problem 18.136P

Students have asked these similar questions

The shutter shown was formed by removing one quarter of a disk of 0.75-in. radius and is used to interrupt a beam of light emanating from a lens at C. Knowing that the shutter weighs 0.125 lb and rotates at the constant rate of 24 cycles per second, determine the magnitude of the force exerted by the shutter on the shaft at A

The blade of a portable saw and the rotor of its motor have a total weight of 2.5 lb and a combined radius of gyration of 1.5 in. Knowing that the blade rotates as shown at the rate w1= 1500 rpm, determine the magnitude and direction of the couple M that a worker must exert on the handle of the saw to rotate it with a constant angular velocity w2= -(2.4 rad/s)j.

A 255-lbf block is suspended from an inextensible cable which is wrapped around a drum of 1.75-ft radius rigidly attached to a flywheel. The drum and flywheel have a combined centroidal moment of inertia 12 lb . ft . s 2 . At the instant shown, the velocity of the block is unknown directed downward. Knowing that the bearing at A is poorly lubricated and that the bearing friction is equivalent to a couple M of magnitude 65 lb .ft, determine the velocity of the block before it has moved 3.5 ft downward if at S2 speed is 13.5ft/s

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 minimum value of θ m .

Solution Summary: The author calculates the principal moment of inertia for principal axes xyz with origin at A.