(a) Explanation Given information: The angular velocity is ω . The angular momentum is H O . The kinetic energy of the rigid body is T . Calculation: The expression for the angular momentum of the rigid body as follows: H O = ( I x ω x − I x y ω y − I x z ω z ) i + ( − I x y ω x + I y ω y − I y z ω z ) j + ( − I x z ω x − I y z ω y + I z ω z ) k The expression for the angular velocity of the rigid body as follows: ω = ω x i + ω y j + ω z k Calculate the dot product of angular momentum and angular velocity using the relation: Dot product = H O ⋅ ω Substitute ( I x ω x − I x y ω y − I x z ω z ) i + ( − I x y ω x + I y ω y − I y z ω z ) j + ( − I x z ω x − I y z ω y + I z ω z ) k for H O and ω x i + ω y j + ω z k for ω . H O ⋅ ω = [ ( I x ω x − I x y ω y − I x z ω z ) i + ( − I x y ω x + I y ω y − I y z ω z ) j + ( − (b) To determine Show that the angle ( θ ) between ω and H O will always be acute.

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

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Chapter 18.1, Problem 18.37P

(a)

To determine

Prove that HO⋅ω=2T.

(b)

To determine

Show that the angle (θ) between ω and HO will always be acute.

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Chapter 18.1, Problem 18.34P

Chapter 18.1, Problem 18.38P

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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...

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Prove that H O ⋅ ω = 2 T .

Solution Summary: The author proves that H_Ocdotomega =2T. The angular momentum and kinetic energy of the rigid body are T.