Concept explainers
The moment of inertia of a system of four particles, about an axis passing through its center of mass and parallel to the z axis.
Answer to Problem 42P
The moment of inertia of a system of four particles, one each at the corners of a square of edge
Explanation of Solution
Given info:
The masses of the balls:
The length of the edge of the cube
The moment of inertia of the system about the z axis
Formula used:
The moment of inertia of a particle of mass
Where,
The moment of inertia of a system of particles about an axis is equal to the sum of moments of inertia of the individual particles about the axis of rotation.
The coordinates
Here,
According to parallel axis theorem,
Here,
Calculation:
Four particles of masses
Figure 1
The mass
Use equations (3) and (4) to determine the coordinates
The center of mass of the system of 4 particles lies at
Determine the distance
Substitute the values of
Determine the value of the mass
Rewrite equation (5) for
Substitute the values of the variables in equation (8) and calculate the moment of inertia of the system of particles about the axis passing through the center of gravity and parallel to the z axis.
Figure 2
The particles of masses
Substitute the values of the given quantities in equation (9) and determine the value of
The calculated value agrees with the value calculated using the parallel axis theorem.
Conclusion:
Thus, the moment of inertia of a system of four particles, one each at the corners of a square of edge
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