Concept explainers
Solve Prob. 18.129, assuming that the meteorite hits the satellite at A instead of B.
The precession angles.
The rate of precession of the satellite after the impact.
The rate of spin axis of the satellite after the impact.
Answer to Problem 18.130P
Thus, the precession angle about the x-axis, y-axis, and z-axis are
The rate of precession of the satellite after impact is
The rate of spin axis of the satellite after impact is
Explanation of Solution
Given information:
The weight of the geostationary satellite is
Write the expression of the mass of the meteorite.
Here, the weight of the meteorite is
Write the expression of the mass of the satellite.
Here, weight of the meteorite is
Write the Expression of the moment of inertia along x-axis.
Here, the radius of gyration of the satellite along x-axis is
Write the Expression of the moment of inertia along the y-axis.
Here, the radius of gyration of the satellite along y-axis is
Write the Expression of the moment of inertia along z-axis.
Here, the radius of gyration of the satellite along z-axis is
Write the expression of the initial momentum of the meteorite.
Here, the initial velocity of the meteorite is
Write the expression of initial angular momentum of the satellite before the impact.
Here, the initial angular velocity along x-axis is
Write the expression of the angular momentum about point A before impact.
Here, the distance between point A, and
Write the expression of angular momentum of the satellite after impact.
Here, the angular velocity after impact along x-axis is
According to the momentum equilibrium, the angular momentum before and after the impact of meteorite at point A will be same.
Write the expression of magnitude net angular velocity after impact of meteorite.
Write the expression of magnitude net angular momentum after impact of meteorite.
The satellite is symmetrical about y-axis.
Write the expression of moment of inertia along the symmetry axis.
Figure-(1) shows the precession axis and the spin axis.
Figure-(1)
Here, angle of precession axis with respect to x, y and z axes is defined in cosine, the angular velocity is
Write the expression of the angle of precession axis with respect to x-axis.
Here, the angle of precession axis with respect to x-axis is
Write the expression of the angle of precession axis with respect to y-axis.
Here, the angle of precession axis with respect to x-axis is
Write the expression of the angle of precession axis with respect to y-axis.
Here, the angle of precession axis with respect to x-axis is
Figure-(2) shows the angle of spin axis with respect to precession axis is constant.
Figure-(2)
Here, the rate of precession is
Write the expression of the angle of precession axis with respect to spin axis is constant from figure (2).
Write the expression of the angle of angular velocity with respect to spin axis.
Here, the angle of angular velocity with respect to spin axis is
Write the expression for relation between
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute in
Substitute
Substitute
Substitute
Equate x-components in Equation (XIX) on both side and substitute
Equate y-components in Equation (XIX) on both side and substitute
Equate z-components in Equation (XIX) on both side and substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Compare first and second term of Equation (XVIII).
Substitute
Compare second and third term of Equation (XVIII).
Substitute
Conclusion:
Thus, the precession angle about the x-axis, y-axis, and z-axis are
The rate of precession of the satellite after impact is
The rate of spin axis of the satellite after impact is
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