Concept explainers
Determine the kinetic energy of the space probe of Prob. 18.34 in its motion about its mass center after its collision with the meteorite.
18.34 The coordinate axes shown represent the principal centroidal axes of inertia of a 3000-lb space probe whose radii of gyration are kx = 1.375 ft, ky = 1.425 ft, and kz = 1.250 ft. The probe has no angular velocity when a 5-oz meteorite strikes one of its solar panels at point A and emerges on the other side of the panel with no change in the direction of its velocity, but with a speed reduced by 25 percent. Knowing that the final angular velocity of the probe is ω = (0.05 rad/s)i − (0.12 rad/s)j + ωzk and that the x component of the resulting change in the velocity of the mass center of the probe is −0.675 in./s, determine (a) the component ωz of the final angular velocity of the probe, (b) the relative velocity v0 with which the meteorite strikes the panel.
Fig. P18.33 and P18.34
The kinetic energy
Answer to Problem 18.54P
The kinetic energy
Explanation of Solution
Given information:
The weight of the space probe
The radius of gyration along x axis
The radius of gyration along y axis
The radius of gyration along z axis
The weight of the meteorite (w) is 5 oz.
The angular velocity
The change in velocity of the mass center of the probe
The width of the side panel from center to point A (b) is 9 ft.
The length of the panel from center to point A (l) is 0.75 ft.
The speed is reduced by 25 percent.
Calculation:
Calculate the mass of the space probe
Here, g is the acceleration due to gravity.
Substitute
Calculate the mass of the meteorite (m) using the formula:
Substitute
Write the relative position vector
Substitute 9 ft for b and 0.75 ft.
Write the expression for the velocity
Calculate the initial liner momentum of the meteorite using the relation:
Substitute
Calculate the moment about origin
Substitute
The speed is reduced to 25 percent.
Calculate the final liner momentum of the meteorite using the relation:
Substitute
Calculate the final linear momentum of meteorite and its moment about the origin using the relation:
Substitute
The initial linear momentum of the space probe
Calculate the final linear momentum of the space probe using the relation:
Substitute
Substitute -0.675 in./s for
Calculate the final angular momentum of the space probe
Substitute
Write the expression for the conservation of linear momentum of the probe plus the meteorite as follows:
Substitute
Equate the i component from the Equation (1).
Equate j component from the Equation (1).
Equate k component from the Equation (1).
Write the expression for the conservation of angular momentum about the origin as follows:
Substitute
Equate i component from the equation (2).
Equate k component from the equation (2).
Substitute –4840 ft/s for
Calculate the kinetic energy of motion of the probe relative to its mass center
Substitute
Thus, the kinetic energy
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Chapter 18 Solutions
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