PEARSON ETEXT ENGINEERING MECH & STATS
PEARSON ETEXT ENGINEERING MECH & STATS
15th Edition
ISBN: 9780137514724
Author: HIBBELER
Publisher: PEARSON
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Chapter 21, 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)
Consider a cylinder of radius R = 0.6 m and radius of gyration k = 0.33 m rolling (without slipping) down an inclined plane. The plane makes an angle of ø = 10 deg with the horizontal plane. How long (in seconds) will it take for the center of cylinder to travel a distance of 1.3 m.
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.
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