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it pass thru centroid , why it is 0 ?HallsofIvy said:
The shape can be thought of as made up of many little areas like A. Each little area has its own (x', y') coordinates relative to the centroid. y-bar is here defined as the average value of y' across all these little areas. By definition of centroid, that average is zero.goldfish9776 said:
why only the second integral = 0 , why not the first integral equal to 0 also ?haruspex said:
Because in the first integral, y' is squared.goldfish9776 said:
The parallel axis theorem for area states that the moment of inertia of a planar shape about an axis parallel to its centroidal axis is equal to the moment of inertia about the centroidal axis plus the product of the shape's area and the square of the distance between the two axes.
The parallel axis theorem is important because it allows us to calculate the moment of inertia for complex shapes that cannot be easily calculated using basic formulas. This is useful in physics and engineering when analyzing the rotational motion and stability of objects.
The parallel axis theorem is derived from the more general parallel axis theorem for moments of inertia, which relates the moment of inertia of a body about any axis to its moment of inertia about a parallel axis through the body's center of mass. This theorem is derived using the basic principles of calculus and geometry.
Yes, the parallel axis theorem for area is limited to planar shapes and cannot be applied to objects with complex 3-dimensional shapes. Additionally, it assumes that the shape is made of a uniform material with a constant density.
No, the parallel axis theorem can only be used for uniform shapes. For non-uniform shapes, the moment of inertia must be calculated using integration methods or through experimental measurements.
