Engineering Mechanics: Statics & Dynamics (14th Edition)
Engineering Mechanics: Statics & Dynamics (14th Edition)
14th Edition
ISBN: 9780133915426
Author: Russell C. Hibbeler
Publisher: PEARSON
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Chapter 21.4, Problem 41P
To determine

Derive the scalar form of the rotational equation of motion about the x axis.

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5. Consider a thin homogeneous plate with principal momenta of inertia along the principal axis x1 along the principal axis x2 I I,> I, I3=I;+I¡ along the principal axis x3 Let the origins of the x; and x; systems coincide and be located at the center of mass 0 of the plate. At time =0, the plate is set rotating in a force-free manner with an angular velocity 2 about an axis inclined at an angle a from the plane of the plate and perpendicular to the x-axis. If I,/I, = cos 2a, show that at time t %3D the angular velocity about the x-axis is w, (1) = N cos a · tanh(2t sin a)
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Consider a satellite moving in a torque-free environment. The inertial frame and satellite body-fixed frame are represented by N-frame and B-frame, where {^₁, ŵ2, ñ3} and {b₁, 62, 63} are right-handed vector bases fixed in N-frame and B-frame, respectively. {1;} is aligned with the principal axis of inertia, where ☎₁ and 3 are associated with the minimum and maximum moments of inertia, respectively. Denote the satellite angular velocity by w = wib; and the inertia tensor about the CoM by Ic. Suppose that I at time t = 0 is given by: Ic(t = 0) = 2ñ₁ñ₁ +2.25 ñ¿Ñ₂ +2.25 ñ³Ñ3 – 0.75 ñ¿ñ3 −0.75 3₂ (dimensionless) Express the satellite inertia tensor about the CoM in B-frame. Determine [BN] at time t = 0, i.c., the direction cosine matrix (DCM) that represents the orientation of B-frame relative to N-frame.

Chapter 21 Solutions

Engineering Mechanics: Statics & Dynamics (14th Edition)

Ch. 21.1 - Prob. 11PCh. 21.1 - Determine the moment of inertia Ixx of the...Ch. 21.1 - Prob. 13PCh. 21.1 - Prob. 14PCh. 21.1 - Prob. 15PCh. 21.1 - Prob. 16PCh. 21.1 - The bent rod has a weight of 1.5 lb/ft. Locate the...Ch. 21.1 - Prob. 18PCh. 21.1 - Prob. 19PCh. 21.1 - Prob. 20PCh. 21.1 - Prob. 21PCh. 21.3 - If a body contains no planes of symmetry, the...Ch. 21.3 - Prob. 23PCh. 21.3 - The 15-kg circular disk spins about its axle with...Ch. 21.3 - Prob. 25PCh. 21.3 - Prob. 26PCh. 21.3 - Prob. 27PCh. 21.3 - Prob. 28PCh. 21.3 - Prob. 29PCh. 21.3 - Prob. 30PCh. 21.3 - Prob. 31PCh. 21.3 - The 2-kg thin disk is connected to the slender rod...Ch. 21.3 - Prob. 33PCh. 21.3 - Prob. 34PCh. 21.3 - The 200-kg satellite has its center of mass at...Ch. 21.3 - Prob. 36PCh. 21.3 - Prob. 37PCh. 21.3 - Determine the kinetic energy of the 7-kg disk and...Ch. 21.3 - Prob. 39PCh. 21.3 - Prob. 40PCh. 21.4 - Prob. 41PCh. 21.4 - Prob. 42PCh. 21.4 - Prob. 43PCh. 21.4 - Prob. 44PCh. 21.4 - Prob. 45PCh. 21.4 - The assembly is supported by journal bearings at A...Ch. 21.4 - Prob. 47PCh. 21.4 - Prob. 48PCh. 21.4 - Prob. 49PCh. 21.4 - Prob. 50PCh. 21.4 - Prob. 51PCh. 21.4 - Prob. 52PCh. 21.4 - Prob. 53PCh. 21.4 - Prob. 54PCh. 21.4 - Prob. 55PCh. 21.4 - Prob. 56PCh. 21.4 - The blades of a wind turbine spin about the shaft...Ch. 21.4 - Prob. 58PCh. 21.4 - The thin rod has a mass of 0.8 kg and a total...Ch. 21.4 - Show that the angular velocity of a body, in terms...Ch. 21.4 - A thin rod is initially coincident with the Z axis...Ch. 21.6 - The gyroscope consists of a uniform 450-g disk D...Ch. 21.6 - The toy gyroscope consists of a rotor R which is...Ch. 21.6 - The top consists of a thin disk that has a weight...Ch. 21.6 - Solve Prob. 2164 when =90.Ch. 21.6 - Prob. 66PCh. 21.6 - Prob. 67PCh. 21.6 - Prob. 68PCh. 21.6 - Prob. 69PCh. 21.6 - Prob. 70PCh. 21.6 - Prob. 71PCh. 21.6 - Prob. 72PCh. 21.6 - Prob. 73PCh. 21.6 - Prob. 74PCh. 21.6 - Prob. 75PCh. 21.6 - Prob. 76PCh. 21.6 - Prob. 77PCh. 21.6 - Prob. 78P
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