Explanation Given information: The weight of the triangular plate ( W ) is 10 lb. The angular velocity ( ω ) is 8 rad/s. The length of AB ( h ) is 24 in. The width of the plate ( b ) is 12 inch. Calculation: Show the diagram of the system as in Figure (1). Write the expression for the angular momentum about point A as follows: H A = − I x y ω i + I y ω j − I y z ω k (1) Here, H A is the angular momentum of about point A , I x y is the angular momentum about A in the xy direction, I y is the angular momentum about A in the y direction and I y z is the angular momentum about A in the yz direction. Write the expression for the product of inertia of area for plate ABC with respect to xy axis ( I x y ) a r e a as follows: ( I x y ) a r e a = 1 24 b 2 h 2 Write the expression for the area of the triangle ABC ( A ) as follows: A = 1 2 b h Calculate the mass moment of inertia in the xy direction ( I x y ) using the formula: I x y = m A ( I x y ) a r e a Substitute 1 24 b 2 h 2 for ( I x y ) a r e a and 1 2 b h for A . I x y = m ( 1 24 b 2 h 2 ) 1 2 b h = 1 12 m b h Write the expression or the product of inertia of area for plate ABC with respect to y axis ( I y ) a r e a as follows: ( I y ) a r e a = 1 12 b 3 h Write the expression for the mass moment of inertia in the y direction I y as follows: I y = m A ( I y ) a r e a Substitute 1 12 b 3 h for ( I y ) a r e a and 1 2 b h for A . I y = 1 12 b 3 h m 1 2 b h = 1 6 b 2 m The z coordinate is negligible. Thus, I y z = 0 Find the angular momentum about point A : Substitute 0 for I y z , 1 6 b 2 m for I y , and 1 12 m b h for I x y in Equation (1). H A = − ( 1 12 m b h ) ω i + ( 1 6 b 2 m ) ω j − ( 0 ) ω k = − 1 12 m b h ω i + 1 6 m b 2 ω j Write the expression for moment about A ( M A ) as follows: M A = h j × ( B x i + B z k ) Here, B x is the force at B in the x direction and B z is the force at B in the z direction. The matrix multiplication for vector product is done. M A = h B z i − h B x k (2) Write the expression for the angular velocity of the reference frame ( Ω ) as follows: Ω = ω j Write the expression for the moment about A as follows: M A = ( H ˙ A ) x y z + Ω × H A Here, ( H ˙ A ) x y z is the angular momentum of the reference frame Axyz . The angular velocity is constant, so the value of ( H ˙ A ) x y z is zero. Substitute 0 for ( H ˙ A ) x y z , ω j for Ω , and − 1 12 m b h ω i + 1 6 m b 2 ω j for H A

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

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 18.2, Problem 18.66P

To determine

The dynamic reactions at A (A) and B (B).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 18.2, Problem 18.65P

Chapter 18.2, Problem 18.67P

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 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 - 18.3 Two uniform rods AB and CE, each of weight 3...Ch. 18.1 - A homogeneous disk of weight W = 6 lb rotates at...Ch. 18.1 - Prob. 18.5P 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 - Prob. 18.11P

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 dynamic reactions at A ( A ) and B ( B ).

Solution Summary: The author explains the dynamic reactions at A ( A ) and B ( B ).

Concept explainers

Videos

The dynamic reactions at A (A) and B (B).

Want to see the full answer?

Chapter 18 Solutions