Question

A circular hoop made of a thin, uniform wire of mass $M$ and radius $R$ has a stationary center-of-mass but exhibits torque-free rotational motion as shown in the sketch. The angular velocity vector has magnitude $omega$ and makes an angle $eta$ with respect to the body $x_3$-axis. a. Compute the principal moments ($I_1, I_2, I_3$) of inertia for the body axes shown in the sketch. b. What is the magnitude of the angular momentum $L$ expressed in the body frame? c. What is the total kinetic energy $T$ expressed in the body coordinate system? d. Compute the precession angular speed $Omega$ of the angular velocity vector $vec{omega}$ around the body symmetric axis $x_3$. e. What is the angle $alpha$ between the body $x_3$-axis and the angular momentum vector? f. Show that $vec{L}$, $vec{omega}$, and the body $x_3$-axis lie in a plan, that is $vec{L} cdot (vec{omega} imes hat{x}_3) = 0$.

          A circular hoop made of a thin, uniform wire of mass $M$ and radius $R$ has a stationary center-of-mass but exhibits torque-free rotational motion as shown in the sketch. The angular velocity vector has magnitude $omega$ and makes an angle $eta$ with respect to the body $x_3$-axis.
a. Compute the principal moments ($I_1, I_2, I_3$) of inertia for the body axes shown in the sketch.
b. What is the magnitude of the angular momentum $L$ expressed in the body frame?
c. What is the total kinetic energy $T$ expressed in the body coordinate system?
d. Compute the precession angular speed $Omega$ of the angular velocity vector $vec{omega}$ around the body symmetric axis $x_3$.
e. What is the angle $alpha$ between the body $x_3$-axis and the angular momentum vector?
f. Show that $vec{L}$, $vec{omega}$, and the body $x_3$-axis lie in a plan, that is $vec{L} cdot (vec{omega} 	imes hat{x}_3) = 0$.

A circular hoop made of a thin, uniform wire of mass M and radius R has a stationary center-of-mass but exhibits torque-free rotational motion as shown in the sketch. The angular velocity vector has magnitude omega and makes an angle eta with respect to the body x3-axis.
a. Compute the principal moments (I1, I2, I3) of inertia for the body axes shown in the sketch.
b. What is the magnitude of the angular momentum L expressed in the body frame?
c. What is the total kinetic energy T expressed in the body coordinate system?
d. Compute the precession angular speed Omega of the angular velocity vector vecomega around the body symmetric axis x3.
e. What is the angle alpha between the body x3-axis and the angular momentum vector?
f. Show that vecL, vecomega, and the body x3-axis lie in a plan, that is vecL cdot (vecomega imes hatx3) = 0.

Added by Cheyenne E.

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